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

Offset

0,3

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

References

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

Links

Table of n, a(n) for n=0..42.

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)

T. 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.

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

Example

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

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

Prog

(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

Crossrefs

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

Cf. A000311.

Sequence in context: A147020 A134685 A130757 * A144006 A014621 A193966

Adjacent sequences: A181993 A181994 A181995 * A181997 A181998 A181999

Keyword

nonn,tabf

Author

N. J. A. Sloane, Apr 05 2012

Extensions

More terms from Michel Marcus, Feb 07 2013

Status

approved