Binomial Ward Numbers

• Binomial Ward numbers are rescaled versions of ordinary Ward numbers, defined by multiplying them with the binomial coefficient (2n choose n+k) and distinguished into two kinds.
• They satisfy triangular recurrence relations that naturally incorporate the binomial factor, illustrating a unique transformation of traditional Ward number recurrences.
• The framework highlights notable limitations, such as the absence of closed forms, generating functions, or combinatorial interpretations, motivating ongoing research in integer sequence theory.

Binomial Ward numbers are two triangular arrays introduced in Section 5 of "Recurrence Relations for Some Integer Sequences Related to Ward Numbers" (Tankosič, 6 Aug 2025). In that paper they are defined as binomially rescaled versions of ordinary Ward numbers of the first and second kind, with the rescaling factor $\binom{2n}{n+k}$. The resulting arrays are denoted $$\left\uparrow\!\begin{matrix} n \ k \end{matrix}\!\right\downarrow^{\circ}$$ and $$\left\updownarrow\!\begin{matrix} n \ k \end{matrix}\!\right\updownarrow^{\circ}$$, are indexed by $n,k\in\mathbb N_0$, typically with $n\ge k$, and are studied through defining transforms and triangular recurrences rather than through explicit closed forms or generating-function theory (Tankosič, 6 Aug 2025).

1. Definition, notation, and boundary conditions

The paper first recalls Luschny’s arrays

$$T_1^\circ(n,k)=(-1)^k\frac{(2n)!}{k!(n-k)!}\,P_n^k\!\left(1,\frac{x}{x+1}\right), \qquad x\in\mathbb N,$$

and

$$T_2^\circ(n,k)=(-1)^k\frac{(2n)!}{k!(n-k)!}\,P_n^k\!\left(\frac{1}{x+1}\right), \qquad x\in\mathbb N_0.$$

In Section 5 these are renamed as

$$T_1^\circ(n,k)= \left\uparrow\!\begin{matrix} n \ k \end{matrix}\!\right\downarrow^{\circ}, \qquad T_2^\circ(n,k)= \left\updownarrow\!\begin{matrix} n \ k \end{matrix}\!\right\updownarrow^{\circ}.$$

Accordingly, the paper distinguishes binomial Ward numbers of the first kind and binomial Ward numbers of the second kind (Tankosič, 6 Aug 2025).

Their boundary conditions are

$$\left\uparrow\!\begin{matrix} n \ 0 \end{matrix}\!\right\downarrow^\circ = \left\uparrow\!\begin{matrix} 0 \ k \end{matrix}\!\right\downarrow^\circ = \left\updownarrow\!\begin{matrix} n \ 0 \end{matrix}\!\right\updownarrow^\circ = \left\updownarrow\!\begin{matrix} 0 \ k \end{matrix}\!\right\updownarrow^\circ =0,$$

together with

$$\left\uparrow\!\begin{matrix} 0 \ 0 \end{matrix}\!\right\downarrow^\circ = \left\updownarrow\!\begin{matrix} 0 \ 0 \end{matrix}\!\right\updownarrow^\circ =1.$$

The arrays are considered for $$\left\uparrow\!\begin{matrix} n \ k \end{matrix}\!\right\downarrow^{\circ}$$0, with effective support typically constrained by $$\left\uparrow\!\begin{matrix} n \ k \end{matrix}\!\right\downarrow^{\circ}$$1 (Tankosič, 6 Aug 2025).

2. Relation to ordinary Ward numbers and the meaning of “binomial”

The defining feature of binomial Ward numbers is that they are obtained from ordinary Ward numbers by multiplication with the binomial factor $$\left\uparrow\!\begin{matrix} n \ k \end{matrix}\!\right\downarrow^{\circ}$$2. For the first kind,

$$\left\uparrow\!\begin{matrix} n \ k \end{matrix}\!\right\downarrow^{\circ}$$3

and for the second kind,

$$\left\uparrow\!\begin{matrix} n \ k \end{matrix}\!\right\downarrow^{\circ}$$4

The paper then remarks: “Since $$\left\uparrow\!\begin{matrix} n \ k \end{matrix}\!\right\downarrow^{\circ}$$5 is the binomial coefficient, we call these numbers binomial Ward numbers” (Tankosič, 6 Aug 2025).

The underlying ordinary Ward numbers of the first kind satisfy, for $$\left\uparrow\!\begin{matrix} n \ k \end{matrix}\!\right\downarrow^{\circ}$$6,

$$\left\uparrow\!\begin{matrix} n \ k \end{matrix}\!\right\downarrow^{\circ}$$7

while ordinary Ward numbers of the second kind satisfy

$$\left\uparrow\!\begin{matrix} n \ k \end{matrix}\!\right\downarrow^{\circ}$$8

They share the same zero-boundary conditions away from $$\left\uparrow\!\begin{matrix} n \ k \end{matrix}\!\right\downarrow^{\circ}$$9 (Tankosič, 6 Aug 2025).

Within the broader terminology of the paper, “binomial” means multiplication by $$\left\updownarrow\!\begin{matrix} n \ k \end{matrix}\!\right\updownarrow^{\circ}$$0, just as “varied” means multiplication by $$\left\updownarrow\!\begin{matrix} n \ k \end{matrix}\!\right\updownarrow^{\circ}$$1. The same pattern reappears later for binomial Ward-Lah numbers: $$\left\updownarrow\!\begin{matrix} n \ k \end{matrix}\!\right\updownarrow^{\circ}$$2 This suggests a systematic binomialization mechanism internal to that 2025 framework (Tankosič, 6 Aug 2025).

3. Triangular recurrences

The main theorems proved specifically for binomial Ward numbers are triangular recurrences. For binomial Ward numbers of the first kind, and for $$\left\updownarrow\!\begin{matrix} n \ k \end{matrix}\!\right\updownarrow^{\circ}$$3,

$$\left\updownarrow\!\begin{matrix} n \ k \end{matrix}\!\right\updownarrow^{\circ}$$4

For binomial Ward numbers of the second kind, again for $$\left\updownarrow\!\begin{matrix} n \ k \end{matrix}\!\right\updownarrow^{\circ}$$5,

$$\left\updownarrow\!\begin{matrix} n \ k \end{matrix}\!\right\updownarrow^{\circ}$$6

These are the paper’s principal structural results for the Section 5 arrays (Tankosič, 6 Aug 2025).

The derivation is entirely by rescaling the ordinary Ward recurrences. In the first-kind case, one starts from

$$\left\updownarrow\!\begin{matrix} n \ k \end{matrix}\!\right\updownarrow^{\circ}$$7

multiplies by

$$\left\updownarrow\!\begin{matrix} n \ k \end{matrix}\!\right\updownarrow^{\circ}$$8

and then identifies shifted terms through

$$\left\updownarrow\!\begin{matrix} n \ k \end{matrix}\!\right\updownarrow^{\circ}$$9

and

$n,k\in\mathbb N_0$0

The second-kind proof follows the same pattern, starting instead from

$n,k\in\mathbb N_0$1

The recurrences therefore encode the binomial prefactor directly into the triangular evolution law (Tankosič, 6 Aug 2025).

4. Equivalent exact formulations and stated limitations

Although Section 5 emphasizes the scaling formulas, the arrays also inherit an implicit partition-transform description from the Luschny notation: $n,k\in\mathbb N_0$2

$n,k\in\mathbb N_0$3

These formulas are exact, but the paper foregrounds their equivalent rescaling by ordinary Ward numbers rather than developing the partition-transform side in detail (Tankosič, 6 Aug 2025).

Equally important is the paper’s explicit negative information. For the Section 5 binomial Ward numbers, it gives no horizontal recurrence, no higher-order recurrence, no explicit closed form, no exponential generating function, no ordinary generating function, no summation formula, no Stirling/Lah-number identity, no matrix or Riordan-array description, no combinatorial interpretation, and no inverse relation. It also does not print a table or list of initial numerical values, even though the introduction points to OEIS entries A268439 and A268440 for Luschny’s arrays $n,k\in\mathbb N_0$4 and $n,k\in\mathbb N_0$5 (Tankosič, 6 Aug 2025).

A further point of clarification concerns Sister Celine’s algorithm. The paper states in its abstract that higher-order recurrences are obtained by using Sister Celine’s general algorithm, but for Section 5 binomial Ward numbers the answer is explicitly negative: no such higher-order recurrence is derived there. A likely typo or mismatch occurs later, when Section 6 says that Sister Celine’s algorithm yields “another recurrence relation for binomial Ward numbers of order 5,” but the theorem that follows is actually written for the notation $n,k\in\mathbb N_0$6, which in that section denotes binomial Ward-Lah numbers, not the Section 5 binomial Ward numbers (Tankosič, 6 Aug 2025).

5. Position within the broader Ward and binomial landscape

The Section 5 arrays are one branch of a broader Ward-number literature in which binomial coefficients enter in several distinct ways. In "On Ward Numbers and Increasing Schröder Trees" (Wang et al., 21 Jul 2025), ordinary Ward numbers $n,k\in\mathbb N_0$7 are defined by

$n,k\in\mathbb N_0$8

and are identified with restricted set partitions by

$n,k\in\mathbb N_0$9

That paper also proves the explicit binomial-sum formula

$n\ge k$0

and, for enriched Ward numbers,

$n\ge k$1

These formulas concern ordinary or enriched Ward numbers rather than the Section 5 binomial Ward numbers, but they show that explicit binomial prefactors and binomial inverse-transform patterns already occur elsewhere in Ward theory (Wang et al., 21 Jul 2025).

A plausible implication is that the Section 5 construction in (Tankosič, 6 Aug 2025) belongs to a wider practice of modifying Ward-type triangles by triangular binomial factors. The paper itself makes this broader pattern explicit only for varied Ward numbers, binomial Ward numbers, and binomial Ward-Lah numbers, but adjacent results on Ward numbers proper reinforce the same theme (Tankosič, 6 Aug 2025).

The structural analogy becomes sharper when compared with other Pascal-type or binomially resolved triangles. "Recursive Harmonic Numbers and Binomial Coefficients" (Maw et al., 2017) studies recursive harmonic numbers $n\ge k$2, organized in a Pascal-like triangular table and satisfying the binomial-sum identity

$n\ge k$3

That paper does not mention Ward numbers, yet it supplies a clear model of how a recursively defined triangle can admit a binomial-coefficient representation. This suggests a methodological comparison point for binomial Ward numbers, especially because the Section 5 arrays of (Tankosič, 6 Aug 2025) are also defined as lower-triangular recurrences with an imposed binomial structure (Maw et al., 2017).

A further adjacent toolkit appears in "Convolutions for Stirling numbers, Lah numbers, and binomial coefficients" (Boyadzhiev, 2021), which develops identities such as

$n\ge k$4

together with mixed Stirling–Lah convolutions. No direct specialization to binomial Ward numbers is given there. Still, this suggests an analytic environment in which further formulas for the Section 5 arrays might eventually be sought, especially if one aims to express them through lower-triangular transforms (Boyadzhiev, 2021).

6. Terminological ambiguity and common misidentifications

The phrase binomial Ward numbers is not uniform across the wider literature, and several nearby objects are distinct from the Section 5 arrays of (Tankosič, 6 Aug 2025). The most important distinction is with the Ward–Horadam binomial framework developed in "A note on Lucas L binomials and Ward Horadam binomials recurrences" (Kwasniewski, 2010) and "Note on Ward-Horadam H(x) - binomials' recurrences and related interpretations, II" (Kwasniewski, 2011). Those papers do not define the Section 5 arrays at all. Instead, they study generalized binomial coefficients attached to second-order Ward–Horadam sequences,

$n\ge k$5

with recurrences induced by addition laws of the underlying sequence. Their subject is therefore generalized $n\ge k$6-binomial coefficients, not binomially rescaled ordinary Ward triangles [(Kwasniewski, 2010); (Kwasniewski, 2011)].

A second source of ambiguity is the Apéry-like sequence

$n\ge k$7

studied in "Congruences involving binomial coefficients and Apéry-like numbers" (Sun, 2018). That paper does not call these values binomial Ward numbers; it treats $n\ge k$8 as an Apéry-like sequence of the second kind and develops congruences for sums such as

$n\ge k$9

Despite the symbol $$T_1^\circ(n,k)=(-1)^k\frac{(2n)!}{k!(n-k)!}\,P_n^k\!\left(1,\frac{x}{x+1}\right), \qquad x\in\mathbb N,$$0, these are unrelated to the Section 5 binomial Ward arrays of (Tankosič, 6 Aug 2025, Sun, 2018).

The most persistent misconception, therefore, is terminological conflation. In current arXiv usage, at least three distinct constructions lie near the phrase: the Section 5 binomial Ward numbers defined by

$$T_1^\circ(n,k)=(-1)^k\frac{(2n)!}{k!(n-k)!}\,P_n^k\!\left(1,\frac{x}{x+1}\right), \qquad x\in\mathbb N,$$1

the Ward–Horadam generalized binomial coefficients $$T_1^\circ(n,k)=(-1)^k\frac{(2n)!}{k!(n-k)!}\,P_n^k\!\left(1,\frac{x}{x+1}\right), \qquad x\in\mathbb N,$$2, and the Apéry-like sequence $$T_1^\circ(n,k)=(-1)^k\frac{(2n)!}{k!(n-k)!}\,P_n^k\!\left(1,\frac{x}{x+1}\right), \qquad x\in\mathbb N,$$3. Only the first of these is the exact object named binomial Ward numbers in (Tankosič, 6 Aug 2025).