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.

1, 0, 1, 0, 4, 3, 0, 15, 60, 15, 0, 56, 700, 840, 105, 0, 210, 6720, 22050, 12600, 945, 0, 792, 58905, 421960, 623700, 207900, 10395, 0, 3003, 492492, 6831825, 20740720, 17342325, 3783780, 135135, 0, 11440, 4012008, 100180080, 551450900, 916515600, 491891400, 75675600, 2027025

Offset: 0

Author

Peter Luschny, Mar 08 2016

Keywords

• nonn
• tabl

Examples

			[1]
[0, 1]
[0, 4, 3]
[0, 15, 60, 15]
[0, 56, 700, 840, 105]
[0, 210, 6720, 22050, 12600, 945]
[0, 792, 58905, 421960, 623700, 207900, 10395]
[0, 3003, 492492, 6831825, 20740720, 17342325, 3783780, 135135]
		

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.

Crossrefs

Cf. A001147, A001791, A268437, A268440, A269939, A269941.

Programs

Maple

# The function PTrans is defined in A269941.
A268439_row := n -> PTrans(n, n->1/(n+1), (n,k) -> (-1)^k*(2*n)!/(k!*(n-k)!)):
seq(print(A268439_row(n)), n=0..8);

Sage

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))
for n in (0..7): print([A268439(n, m) for m in (0..n)])

Sage

# uses[PtransMatrix from A269941]
# Alternatively
PtransMatrix(8, lambda n: 1/(n+1), lambda n,k: (-1)^k*factorial(2*n)/ (factorial(k)*factorial(n-k)))

Formula

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.

T(n,k) = A269939(n,k)*binomial(2*n,n+k).

T(n,k) = A268437(n,k)/(k!*(n-k)!).

T(n,1) = binomial(2*n,n-1) = A001791(n) for n>=1.

T(n,n) = (2*n-1)!! = A001147(n) for n>=0.