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A054654 Triangle read by rows: matrix product of the binomial coefficients with the Stirling numbers of the first kind. 11
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, 1, -28, 322, -1960, 6769, -13132, 13068, -5040, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

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 Deléham, Dec 30 2006

From Doudou Kisabaka (dougk7(AT)gmail.com), Dec 18 2009: (Start)

The sum of the entries on each row of the triangle, starting on the 3rd row, equals 0. E.g., 1+(-3)+2+0 = 0.

The entries on the triangle can be computed as follows. T(n,r) = T(n-1,r) - (n-1)*T(n-1,r-1). T(n,r) = 0 when r equals 0 or r > n. T(n,r) = 1 if n==1. (End)

LINKS

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

Eric Weisstein's World of Mathematics, Pochhammer Symbol

Eric Weisstein's World of Mathematics, Rising Factorial

Eric Weisstein's World of Mathematics, FallingFactorial

FORMULA

n!*binomial(x, n) = Sum_{k=0..n} T(n, k)*x^(n-k).

(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.

EXAMPLE

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;

  ...

MAPLE

a054654_row := proc(n) local k; seq(coeff(expand((-1)^n*pochhammer (-x, n)), x, n-k), k=0..n) end:

MATHEMATICA

row[n_] := Reverse[ CoefficientList[ (-1)^n*Pochhammer[-x, n], x] ]; Flatten[ Table[ row[n], {n, 0, 8}]] (* Jean-François Alcover, Feb 16 2012, after Maple *)

PROG

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

(Haskell)

a054654 n k = a054654_tabl !! n !! k

a054654_row n = a054654_tabl !! n

a054654_tabl = map reverse a048994_tabl

-- Reinhard Zumkeller, Mar 18 2014

CROSSREFS

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

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

Sequence in context: A081576 A330785 A292717 * A253669 A154477 A322324

Adjacent sequences:  A054651 A054652 A054653 * A054655 A054656 A054657

KEYWORD

tabl,sign,easy,nice

AUTHOR

N. J. A. Sloane, Apr 18 2000

EXTENSIONS

Additional comments from Thomas Wieder, Dec 29 2006

Edited by N. J. A. Sloane at the suggestion of Eric W. Weisstein, Jan 20 2008

STATUS

approved

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Last modified May 24 19:14 EDT 2020. Contains 334580 sequences. (Running on oeis4.)