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
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)
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
KEYWORD
nonn,tabf
AUTHOR
N. J. A. Sloane, Apr 05 2012
EXTENSIONS
More terms from Michel Marcus, Feb 07 2013
STATUS
approved