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A181996
Triangle of Ward numbers T(n,k) (n>=0, k=0 if n=0, otherwise 0 <= k <= n-1) read by rows.
4
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
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)
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
KEYWORD
nonn,tabf
AUTHOR
N. J. A. Sloane, Apr 05 2012
EXTENSIONS
More terms from Michel Marcus, Feb 07 2013
STATUS
approved