A054654 Triangle of Stirling numbers of 1st kind, S(n, n-k), n >= 0, 0 <= k <= n.

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

Offset

0,8

Comments

Triangle is the matrix product of the binomial coefficients with the Stirling numbers of the first kind.

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, 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.

T(n, k) = (-1)^k*Sum_{j=0..k} E2(k, j)*binomial(n+j-1, 2*k), where E2(k, j) are the second-order Eulerian numbers A340556. - Peter Luschny, Feb 21 2021

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: # Peter Luschny, Nov 28 2010
seq(seq(Stirling1(n, n-k), k=0..n), n=0..8); # Peter Luschny, Feb 21 2021

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 *)
Table[StirlingS1[n, n-k], {n, 0, 10}, {k, 0, n}]//Flatten (* Harvey P. Dale, Jun 17 2023 *)

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.

The Stirling2 counterpart is A106800.

Cf. A054655, A039810, A039814, A126350, A126351, A126353, A340556.

Sequence in context: A330785 A292717 A365727 * A355423 A345066 A253669

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