A268440 Triangle read by rows, T(n,k) = C(2*n,n+k)*Sum_{m=0..k} (-1)^(m+k)*C(n+k,n+m)* Stirling1(n+m,m), for n>=0 and 0<=k<=n.
1, 0, 1, 0, 8, 3, 0, 90, 120, 15, 0, 1344, 3640, 1680, 105, 0, 25200, 110880, 107100, 25200, 945, 0, 570240, 3617460, 5815040, 2910600, 415800, 10395, 0, 15135120, 128576448, 303963660, 256736480, 78828750, 7567560, 135135
Offset: 0
Examples
[1] [0, 1] [0, 8, 3] [0, 90, 120, 15] [0, 1344, 3640, 1680, 105] [0, 25200, 110880, 107100, 25200, 945] [0, 570240, 3617460, 5815040, 2910600, 415800, 10395]
Links
- Peter Luschny, The P-transform.
- Aleks Žigon Tankosič, Recurrence Relations for Some Integer Sequences Related to Ward Numbers, arXiv:2508.04754 [math.CO], 2025. See p. 3.
Programs
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Maple
# The function PTrans is defined in A269941. A268440_row := n -> PTrans(n, n->n/(n+1), (n,k) -> (-1)^k*(2*n)!/(k!*(n-k)!)): seq(print(A268440_row(n)), n=0..8);
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Sage
A268440 = lambda n, k: binomial(2*n,n+k)*sum((-1)^(m+k)*binomial(n+k,n+m)* stirling_number1(n+m, m) for m in (0..k)) for n in (0..7): print([A268440(n, m) for m in (0..n)])
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Sage
# uses[PtransMatrix from A269941] # Alternatively PtransMatrix(7, lambda n: n/(n+1), lambda n,k: (-1)^k*factorial(2*n)/ (factorial(k)*factorial(n-k)))