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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. 2
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

LINKS

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

Peter Luschny, The P-transform.

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

(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)]

# Alternatively with the function PtransMatrix which is defined in A269941:

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: A010521 A200025 A200300 * A137433 A318409 A119278

Adjacent sequences:  A268437 A268438 A268439 * A268441 A268442 A268443

KEYWORD

nonn,tabl

AUTHOR

Peter Luschny, Mar 08 2016

STATUS

approved

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Last modified September 20 03:32 EDT 2019. Contains 327209 sequences. (Running on oeis4.)