A054654 - OEIS

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

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

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)

REFERENCES

Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 18, table 18:6:1 at page 152.

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, 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.