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

0,5

Links

Table of n, a(n) for n=0..35.

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.

Formula

T(n,k) = ((-1)^k*(2*n)!/(k!*(n-k)!))*P[n,k](n/(n+1)) where P is the P-transform. The P-transform is defined in the link.

T(n,k) = A269940*binomial(2*n,n+k).

T(n,k) = A268438(n,k)/(k!*(n-k)!).

T(n,1) = n*(2*n)!/(n+1)! for n>=1 (cf. A092956).

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

Example

[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]

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);

Prog

(SageMath)
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)])
(SageMath) # 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)))

Crossrefs

Cf. A001147, A092956, A268438, A268439, A269940, A269941.

Sequence in context: A200025 A372585 A200300 * A137433 A318409 A119278

Adjacent sequences: A268437 A268438 A268439 * A268441 A268442 A268443

Keyword

nonn,tabl

Author

Peter Luschny, Mar 08 2016

Status

approved