This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A269940 #57 Aug 15 2025 12:57:22
%S A269940 1,0,1,0,2,3,0,6,20,15,0,24,130,210,105,0,120,924,2380,2520,945,0,720,
%T A269940 7308,26432,44100,34650,10395,0,5040,64224,303660,705320,866250,
%U A269940 540540,135135,0,40320,623376,3678840,11098780,18858840,18288270,9459450,2027025
%N A269940 Triangle read by rows, T(n, k) = Sum_{m=0..k} (-1)^(m + k)*binomial(n + k, n + m) * |Stirling1(n + m, m)|, for n >= 0, 0 <= k <= n.
%C A269940 We propose to call this sequence the 'Ward cycle numbers' and sequence A269939 the 'Ward set numbers'. - _Peter Luschny_, Nov 25 2022
%H A269940 Bishal Deb and Alan D. Sokal, <a href="https://arxiv.org/abs/2507.18959">Higher-order Stirling cycle and subset triangles: Total positivity, continued fractions and real-rootedness</a>, arXiv:2507.18959 [math.CO], 2025. See p. 6.
%H A269940 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/P-Transform">The P-transform</a>.
%H A269940 Anthony Mansuy, <a href="https://doi.org/10.1016/j.jalgebra.2014.04.017">Preordered forests, packed words and contraction algebras</a>, J. Algebra 411 (2014) 259-311, section 4.4.
%H A269940 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 A269940 Nico M. Temme, <a href="https://doi.org/10.1080/10652469.2021.1886094">Asymptotic expansions of Kummer hypergeometric functions for large values of the parameters</a>, Integral Transforms and Special Functions, 2021.
%H A269940 Nico M. Temme and Ed J. M. Veling, <a href="https://arxiv.org/abs/2202.12857">Asymptotic expansions of Kummer hypergeometric functions with three asymptotic parameters a, b and z</a>, arXiv:2202.12857 [math.CA], 2022.
%H A269940 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. 12.
%F A269940 T(n,k) = (-1)^k*FF(n+k,n)*P[n,k](n/(n+1)) where P is the P-transform and FF the falling factorial function. For the definition of the P-transform see the link.
%F A269940 T(n,k) = A268438(n,k)*FF(n+k,n)/(2*n)!.
%e A269940 Triangle T(n,k) starts:
%e A269940 [1]
%e A269940 [0, 1]
%e A269940 [0, 2, 3]
%e A269940 [0, 6, 20, 15]
%e A269940 [0, 24, 130, 210, 105]
%e A269940 [0, 120, 924, 2380, 2520, 945]
%e A269940 [0, 720, 7308, 26432, 44100, 34650, 10395]
%e A269940 [0, 5040, 64224, 303660, 705320, 866250, 540540, 135135]
%p A269940 T := (n, k) -> add((-1)^(m+k)*binomial(n+k,n+m)*abs(Stirling1(n+m, m)), m=0..k):
%p A269940 seq(print(seq(T(n, k), k=0..n)), n=0..6);
%p A269940 # Alternatively:
%p A269940 T := proc(n, k) option remember;
%p A269940 `if`(k=0, k^n,
%p A269940 `if`(k<=0 or k>n, 0,
%p A269940 (n+k-1)*(T(n-1, k)+T(n-1, k-1)))) end:
%p A269940 for n from 0 to 6 do seq(T(n, k), k=0..n) od;
%t A269940 T[n_, k_] := Sum[(-1)^(m+k)*Binomial[n+k, n+m]*Abs[StirlingS1[n+m, m]], {m, 0, k}];
%t A269940 Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Sep 12 2022 *)
%o A269940 (Sage)
%o A269940 T = lambda n, k: sum((-1)^(m+k)*binomial(n+k, n+m)*stirling_number1(n+m, m) for m in (0..k))
%o A269940 for n in (0..7): print([T(n, k) for k in (0..n)])
%o A269940 (Sage) # uses[PtransMatrix from A269941]
%o A269940 PtransMatrix(8, lambda n: n/(n+1), lambda n, k: (-1)^k*falling_factorial(n+k,n))
%Y A269940 Variants: A111999, A259456.
%Y A269940 Cf. A269939 (Stirling2 counterpart), A268438, A032188 (row sums).
%K A269940 nonn,tabl
%O A269940 0,5
%A A269940 _Peter Luschny_, Mar 27 2016
%E A269940 Name corrected after notice from Ed Veling by _Peter Luschny_, Jun 14 2022