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A094645 Generalized Stirling number triangle of first kind. 10
1, -1, 1, 0, -1, 1, 0, -1, 0, 1, 0, -2, -1, 2, 1, 0, -6, -5, 5, 5, 1, 0, -24, -26, 15, 25, 9, 1, 0, -120, -154, 49, 140, 70, 14, 1, 0, -720, -1044, 140, 889, 560, 154, 20, 1, 0, -5040, -8028, -64, 6363, 4809, 1638, 294, 27, 1, 0, -40320, -69264, -8540, 50840, 44835, 17913, 3990, 510, 35, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,12

COMMENTS

From Wolfdieter Lang, Jun 20 2011: (Start)

The row polynomials s(n,x):=sum(T(n,k)*x^k,k=0..n) satisfy risefac(x-1,n)=s(n,x), with the rising factorials risefac(x-1,n):=product(x-1+j,j=0..n-1), n>=1, risefac(x-1,0)=1. Compare with the formula risefac(x,n)=s1(n,x), with the row polynomials s1(n,x) of A132393 (unsigned Stirling1).

This is the lower triangular Sheffer array with e.g.f.

  T(x,z) = (1-z)*exp(-x*log(1-z)) (the rewritten e.g.f. from the formula section). See the W. Lang link under A006232 for Sheffer matrices and the Roman reference. In the notation which indicates the column e.g.f.s this is Sheffer (1-z,-log(1-z)). In the umbral notation (cf. Roman) this is called Sheffer for (exp(t),1-exp(-t)).

The row polynomials satisfy s(n,x) = (x+n-1)*s(n-1,x), s(0,x)=1, and s(n,x) = (x-1)*s1(n-1,x), n>=1, s1(0,x)=1, with the unsigned Stirling1 row polynomials s1(n,x).

The row polynomials satisfy also

s(n,x) - s(n,x-1) = n*s(n-1,x), n>1, s(0,x)=1.

  (from the Meixner identity, see the Meixner reference given under A060338).

The row polynomials satisfy as well (from corollary 3.7.2. p. 50 of the Roman reference)

  s(n,x) = (x-1)*s(n-1,x+1), n>=1, s(0,n)=1.

The exponential convolution identity is

  s(n,x+y) = sum(binomial(n,k)*s(k,y)*s1(n-k,x),k=0..n),

  n>=0, with symmetry x <-> y.

The row sums are 1 for n=0 and 0 else, and the alternating  row sums are 1,-2,2, followed by zeros, with e.g.f.(1-x)^2.

The Sheffer a-sequence Sha(n)=A164555(n)/A027642(n) with e.g.f. x/(1-exp(-x)), and the z-sequence is Shz(n)=-1 with e.g.f. -exp(x).

The inverse Sheffer matrix is ((-1)^(n-k))*A105794(n,k) with e.g.f. exp(z)*exp(x*(1-exp(-z))). (End)

Triangle T(n,k), read by rows, given by (-1, 1, 0, 2, 1, 3, 2, 4, 3, 5, ...) DELTA (1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 16 2012

Also coefficients of t in t*(t-1)*Sum[(-1)^(n+m) t^(m-1) StirlingS1[n,m], {m,n}] in which setting t^k equal to k gives n!, from this follows that the dot product of row n with [0,..,n] equals (n-1)!. - Wouter Meeussen, May 15 2012

REFERENCES

S. Roman, The Umbral Calculus, Academic Press, New York, 1984.

LINKS

Table of n, a(n) for n=0..65.

M. W. Coffey, M. C. Lettington, On Fibonacci Polynomial Expressions for Sums of mth Powers, their implications for Faulhaber's Formula and some Theorems of Fermat, arXiv:1510.05402 [math.NT], 2015.

FORMULA

E.g.f.: (1-y)^(1-x).

Sum_{k, 0<=k<=n}T(n,k)*x^k = A000007(n), A000142(n), A000142(n+1), A001710(n+2), A001715(n+3), A001720(n+4), A001725(n+5), A001730(n+6), A049388(n), A049389(n), A049398(n), A051431(n) for x = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 respectively. - Philippe Deléham, Nov 13 2007

If we define f(n,i,a)=sum(binomial(n,k)*stirling1(n-k,i)*product(-a-j,j=0..k-1),k=0..n-i), then |T(n,i)| =|f(n,i,-1)|, for n=1,2,...;i=0...n. - Milan Janjic, Dec 21 2008

From Wolfdieter Lang, Jun 20 2011: (Start)

T(n,k) = |S1(n-1,k-1)| - |S1(n-1,k)|, n>=1, k>=1, with |S1(n,k)|= A132393(n,k) (unsigned Stirling1).

Recurrence: T(n,k) = T(n-1,k-1) +(n-2)*T(n-1,k) if n>=k>=0; T(n,k)=0 if n<k; T(n,-1)=0; T(0,0)=1.

E.g.f. column k: (1-x)*((-log(1-x))^k)/k!. (End)

EXAMPLE

1;

-1,  1;

0, -1,  1;

0, -1,  0,  1;

0, -2, -1,  2,  1;

0, -6, -5,  5,  5, 1;

0,-24,-26, 15, 25, 9, 1;

...

Recurrence:

-2 = T(4,1) = T(3,0) + (4-2)*T(3,1) = 0 + 2*(-1).

Row polynomials:

s(3,x) = -x+x^3 = (x-1)*s1(2,x) = (x-1)*(x+x^2).

s(3,x) = (x-1)*s(2,x+1) = (x-1)*(-(x+1)+(x+1)^2).

s(3,x) - s(3,x-1) = -x+x^3 -(-(x-1)+(x-1)^3) = 3*(-x+x^2) = 3*s(2,x).

MAPLE

A094645_row := n -> seq((-1)^(n-k)*coeff(expand(pochhammer(x-n+2, n)), x, k), k=0..n): seq(print(A094645_row(n)), n=0..6); # Peter Luschny, May 16 2013

MATHEMATICA

t[n_, k_] /; n >= k >= 0 := t[n, k] = t[n-1, k-1] + (n-2)*t[n-1, k]; t[n_, k_] /; n < k = 0; t[_, -1] = 0; t[0, 0] = 1; Flatten[ Table[ t[n, k], {n, 0, 10}, {k, 0, n}] ] (* Jean-François Alcover, Sep 29 2011, after recurrence *);

Table[CoefficientList[t*(t-1)*Sum[(-1)^(n+m)*t^(m-1)*StirlingS1[n, m], {m, n}], t], {n, 1, 7}] (* Wouter Meeussen, May 15 2012 *)

CROSSREFS

Cf. A049444, A049458, A094646, A132393, A105794.

Sequence in context: A240159 A269942 * A105793 A158566 A128410 A059782

Adjacent sequences:  A094642 A094643 A094644 * A094646 A094647 A094648

KEYWORD

easy,sign,tabl

AUTHOR

Vladeta Jovovic, May 17 2004

STATUS

approved

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Last modified May 23 23:53 EDT 2017. Contains 286937 sequences.