A181996 Triangle of Ward numbers T(n,k) (n>=0, k=0 if n=0, otherwise 0 <= k <= n-1) read by rows.

1, 1, 3, 1, 15, 10, 1, 105, 105, 25, 1, 945, 1260, 490, 56, 1, 10395, 17325, 9450, 1918, 119, 1, 135135, 270270, 190575, 56980, 6825, 246, 1, 2027025, 4729725, 4099095, 1636635, 302995, 22935, 501, 1, 34459425, 91891800, 94594500, 47507460, 12122110, 1487200, 74316, 1012, 1

Offset: 0

Author

N. J. A. Sloane, Apr 05 2012

Keywords

• nonn
• tabf

Comments

It appears that the sum of row(n) is A000311(n+1). - Michel Marcus, Feb 07 2013

Conjecture on row sums was proved in the first paragraph of the formula section of the reverse matrix A134991 in 2008 (e.g.f. evaluated at t=1). - Tom Copeland, Jan 03 2016

Examples

			Triangle begins:
      1
      1
      3     1
     15    10    1
    105   105   25    1
    945  1260  490   56   1
  10395 17325 9450 1918 119 1 ...
		

References

• Charles Jordan, Calculus of Finite Differences, Chelsea 1950, p. 172, Table C_{m, i}.

Links

• L. Carlitz, The coefficients in an asymptotic expansion and certain related numbers, Duke Math. J., vol 35 (1968), p. 83-90. See page 85.
• José L. Cereceda, Figurate numbers and sums of powers of integers, arXiv:2001.03208 [math.NT], 2020. See Table 8 p. 11.
• Lane Clark, Asymptotic normality of the Ward numbers, Discrete Math. 203 (1999), no. 1-3, 41-48. MR1696232 (2000d:11101)
• Tom Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms
• Andrew Elvey Price and Alan D. Sokal, Phylogenetic trees, augmented perfect matchings, and a Thron-type continued fraction (T-fraction) for the Ward polynomials, arXiv:2001.01468 [math.CO], 2020.
• Aleks Žigon Tankosič, Recurrence Relations for Some Integer Sequences Related to Ward Numbers, arXiv:2508.04754 [math.CO], 2025. See p. 2.
• Elena L. Wang and Guoce Xin, On Ward Numbers and Increasing Schröder Trees, arXiv:2507.15654 [math.CO], 2025. See p. 1.

Crossrefs

See A134991, which is the mirror image and is the main entry for this triangle, for further information.

Cf. A000311.

Programs

Maple

A181996 := (n,k) -> add((-1)^(n - k + m)*binomial(2*n - k, n + m)*Stirling2(n + m, m), m = 0..n-k):
seq(seq(A181996(n, k), k = 0..n-1+0^n), n=0..8); # Peter Luschny, Feb 19 2021

PARI

T(n,k) = {if (n == 0, return(1)); if (k == 0, return (prod(x=2,n, 2*x-1))); if (k == n, return (0)); return((2*n-1-k)*T(n-1,k) + (n-k)*T(n-1, k-1));} \\ Michel Marcus, Feb 07 2013

Formula

T(n, k) = Sum_{m = 0..n-k} (-1)^(n - k + m)*C(2*n - k, n + m)*Stirling2(n + m, m). - Peter Luschny, Feb 19 2021

Extensions

More terms from Michel Marcus, Feb 07 2013