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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.

%I A181996 #68 Aug 15 2025 10:33:51
%S A181996 1,1,3,1,15,10,1,105,105,25,1,945,1260,490,56,1,10395,17325,9450,1918,
%T A181996 119,1,135135,270270,190575,56980,6825,246,1,2027025,4729725,4099095,
%U A181996 1636635,302995,22935,501,1,34459425,91891800,94594500,47507460,12122110,1487200,74316,1012,1
%N A181996 Triangle of Ward numbers T(n,k) (n>=0, k=0 if n=0, otherwise 0 <= k <= n-1) read by rows.
%C A181996 It appears that the sum of row(n) is A000311(n+1). - _Michel Marcus_, Feb 07 2013
%C A181996 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
%D A181996 Charles Jordan, Calculus of Finite Differences, Chelsea 1950, p. 172, Table C_{m, i}.
%H A181996 L. Carlitz, <a href="http://projecteuclid.org/euclid.dmj/1077377459">The coefficients in an asymptotic expansion and certain related numbers</a>, Duke Math. J., vol 35 (1968), p. 83-90. See page 85.
%H A181996 José L. Cereceda, <a href="https://arxiv.org/abs/2001.03208">Figurate numbers and sums of powers of integers</a>, arXiv:2001.03208 [math.NT], 2020. See Table 8 p. 11.
%H A181996 Lane Clark, <a href="http://dx.doi.org/10.1016/S0012-365X(99)00019-9">Asymptotic normality of the Ward numbers</a>, Discrete Math. 203 (1999), no. 1-3, 41-48. MR1696232 (2000d:11101)
%H A181996 Tom Copeland, <a href="http://tcjpn.wordpress.com/2015/12/21/generators-inversion-and-matrix-binomial-and-integral-transforms/">Generators, Inversion, and Matrix, Binomial, and Integral Transforms</a>
%H A181996 Andrew Elvey Price and Alan D. Sokal, <a href="https://arxiv.org/abs/2001.01468">Phylogenetic trees, augmented perfect matchings, and a Thron-type continued fraction (T-fraction) for the Ward polynomials</a>, arXiv:2001.01468 [math.CO], 2020.
%H A181996 Aleks Žigon Tankosič, <a href="https://arxiv.org/abs/2508.04754">Recurrence Relations for Some Integer Sequences Related to Ward Numbers</a>, arXiv:2508.04754 [math.CO], 2025. See p. 2.
%H A181996 Elena L. Wang and Guoce Xin, <a href="https://arxiv.org/abs/2507.15654">On Ward Numbers and Increasing Schröder Trees</a>, arXiv:2507.15654 [math.CO], 2025. See p. 1.
%F A181996 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
%e A181996 Triangle begins:
%e A181996       1
%e A181996       1
%e A181996       3     1
%e A181996      15    10    1
%e A181996     105   105   25    1
%e A181996     945  1260  490   56   1
%e A181996   10395 17325 9450 1918 119 1 ...
%p A181996 A181996 := (n,k) -> add((-1)^(n - k + m)*binomial(2*n - k, n + m)*Stirling2(n + m, m), m = 0..n-k):
%p A181996 seq(seq(A181996(n, k), k = 0..n-1+0^n), n=0..8); # _Peter Luschny_, Feb 19 2021
%o A181996 (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
%Y A181996 See A134991, which is the mirror image and is the main entry for this triangle, for further information.
%Y A181996 Cf. A000311.
%K A181996 nonn,tabf
%O A181996 0,3
%A A181996 _N. J. A. Sloane_, Apr 05 2012
%E A181996 More terms from _Michel Marcus_, Feb 07 2013