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A054655 Triangle T(n,k) giving coefficients in expansion of n!*binomial(x-n,n) in powers of x. 5
1, 1, -1, 1, -5, 6, 1, -12, 47, -60, 1, -22, 179, -638, 840, 1, -35, 485, -3325, 11274, -15120, 1, -51, 1075, -11985, 74524, -245004, 332640, 1, -70, 2086, -34300, 336049, -1961470, 6314664, -8648640, 1, -92, 3682, -83720, 1182769 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

LINKS

Robert Israel, Table of n, a(n) for n = 0..10010 (rows 0 to 140, flattened).

FORMULA

n!*binomial(x-n, n)= Sum T(n, k)*x^(n-k), k=0..n.

From Robert Israel, Jul 12 2016: (Start)

G.f. Sum_{n>=0} Sum_{k=0..n} T(n,k)*x^n*y^k  = hypergeom([1, -1/(2*y), (1/2)*(-1+y)/y], [-1/y], -4*x*y).

E.g.f. Sum_{n>=0} Sum_{k=0..n} T(n,k)*x^n*y^k/n! = (1+4*x*y)^(-1/2)*((1+sqrt(1+4*x*y))/2)^(1+1/y). (End)

EXAMPLE

1; 1,-1; 1,-5,6; 1,-12,47,-60; ...

MAPLE

a054655_row := proc(n) local k; seq(coeff(expand((-1)^n*pochhammer (n-x, n)), x, n-k), k=0..n) end: # Peter Luschny, Nov 28 2010

MATHEMATICA

row[n_] := Table[ Coefficient[(-1)^n*Pochhammer[n - x, n], x, n - k], {k, 0, n}]; A054655 = Flatten[ Table[ row[n], {n, 0, 8}]] (* Jean-Fran├žois Alcover, Apr 06 2012, after Maple *)

PROG

(PARI) T(n, k)=polcoeff(n!*binomial(x-n, n), n-k)

CROSSREFS

Cf. A054651, A054654, A008276.

Sequence in context: A113262 A195823 A105577 * A290319 A086745 A086646

Adjacent sequences:  A054652 A054653 A054654 * A054656 A054657 A054658

KEYWORD

sign,tabl,easy,nice

AUTHOR

N. J. A. Sloane, Apr 18 2000

STATUS

approved

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Last modified February 17 17:27 EST 2018. Contains 299296 sequences. (Running on oeis4.)