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A268439 Triangle read by rows, T(n,k) = C(2*n,n+k)*Sum_{m=0..k} (-1)^(m+k)*C(n+k,n+m)* Stirling2(n+m,m), for n>=0 and 0<=k<=n.

%I A268439 #25 Aug 15 2025 10:34:02
%S A268439 1,0,1,0,4,3,0,15,60,15,0,56,700,840,105,0,210,6720,22050,12600,945,0,
%T A268439 792,58905,421960,623700,207900,10395,0,3003,492492,6831825,20740720,
%U A268439 17342325,3783780,135135,0,11440,4012008,100180080,551450900,916515600,491891400,75675600,2027025
%N A268439 Triangle read by rows, T(n,k) = C(2*n,n+k)*Sum_{m=0..k} (-1)^(m+k)*C(n+k,n+m)* Stirling2(n+m,m), for n>=0 and 0<=k<=n.
%H A268439 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/P-Transform">The P-transform</a>.
%H A268439 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. 3.
%F A268439 T(n,k) = ((-1)^k*(2*n)!/(k!*(n-k)!))*P[n,k](1/(n+1)) where P is the P-transform. The P-transform is defined in the link.
%F A268439 T(n,k) = A269939(n,k)*binomial(2*n,n+k).
%F A268439 T(n,k) = A268437(n,k)/(k!*(n-k)!).
%F A268439 T(n,1) = binomial(2*n,n-1) = A001791(n) for n>=1.
%F A268439 T(n,n) = (2*n-1)!! = A001147(n) for n>=0.
%e A268439 [1]
%e A268439 [0, 1]
%e A268439 [0, 4, 3]
%e A268439 [0, 15, 60, 15]
%e A268439 [0, 56, 700, 840, 105]
%e A268439 [0, 210, 6720, 22050, 12600, 945]
%e A268439 [0, 792, 58905, 421960, 623700, 207900, 10395]
%e A268439 [0, 3003, 492492, 6831825, 20740720, 17342325, 3783780, 135135]
%p A268439 # The function PTrans is defined in A269941.
%p A268439 A268439_row := n -> PTrans(n, n->1/(n+1), (n,k) -> (-1)^k*(2*n)!/(k!*(n-k)!)):
%p A268439 seq(print(A268439_row(n)), n=0..8);
%o A268439 (Sage)
%o A268439 A268439 = lambda n, k: binomial(2*n, n+k)*sum((-1)^(m+k)*binomial(n+k, n+m)* stirling_number2(n+m, m) for m in (0..k))
%o A268439 for n in (0..7): print([A268439(n, m) for m in (0..n)])
%o A268439 (Sage) # uses[PtransMatrix from A269941]
%o A268439 # Alternatively
%o A268439 PtransMatrix(8, lambda n: 1/(n+1), lambda n,k: (-1)^k*factorial(2*n)/ (factorial(k)*factorial(n-k)))
%Y A268439 Cf. A001147, A001791, A268437, A268440, A269939, A269941.
%K A268439 nonn,tabl
%O A268439 0,5
%A A268439 _Peter Luschny_, Mar 08 2016