0,8

Triangle T(n,k) giving coefficients in expansion of n!*C(x,n) in powers of x. E.g. 3!*C(x,3) = x^3-3*x^2+2*x.

The matrix product of binomial coefficients with the Stirling numbers of the first kind results in the Stirling numbers of the first kind again, but the triangle is shifted by (1,1).

Essentially [1,0,1,0,1,0,1,0,...] DELTA [0,-1,-1,-2,-2,-3,-3,-4,-4,...] where DELTA is the operator defined in A084938 ; mirror image of the Stirling-1 triangle A048994 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 30 2006

Eric Weisstein's World of Mathematics, Pochhammer Symbol

Eric Weisstein's World of Mathematics, Rising Factorial

Eric Weisstein's World of Mathematics, FallingFactorial

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

(In Maple notation:) Matrix product A*B of matrix A[i,j]:=binomial(j-1,i-1) with i = 1 to p+1, j = 1 to p+1, p=8 and of matrix B[i,j]:=stirling1(j,i) with i from 1 to d, j from 1 to d, d=9.

Matrix begins:

1 0 0 0 0 0 0 0 0

0 1 -1 2 -6 24 -120 720 -5040

0 0 1 -3 11 -50 274 -1764 13068

0 0 0 1 -6 35 -225 1624 -13132

0 0 0 0 1 -10 85 -735 6769

0 0 0 0 0 1 -15 175 -1960

0 0 0 0 0 0 1 -21 322

0 0 0 0 0 0 0 1 -28

0 0 0 0 0 0 0 0 1

...

Triangle begins:

1;

1, 0;

1, -1, 0;

1, -3, 2, 0;

1, -6, 11, -6, 0;

1, -10, 35, -50, 24, 0;

1, -15, 85, -225, 274, -120, 0;

1, -21, 175, -735, 1624, -1764, 720, 0;

...

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

Essentially Stirling numbers of first kind, multiplied by factorials - see A008276. Cf. A054655.

Cf. A039810, A039814, A126350, A126351, A126353.

Sequence in context: A008783 A139144 A081576 this_sequence A154477 A142071 A118972

Adjacent sequences: A054651 A054652 A054653 this_sequence A054655 A054656 A054657

tabl,sign,easy,nice

N. J. A. Sloane (njas(AT)research.att.com), Apr 18 2000

Additional comments from Thomas Wieder (thomas.wieder(AT)t-online.de), Dec 29 2006

Edited by N. J. A. Sloane (njas(AT)research.att.com) at the suggestion of Eric Weisstein, Jan 20 2008