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 A094645 Generalized Stirling number triangle of first kind. 10

%I

%S 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,

%T 0,-120,-154,49,140,70,14,1,0,-720,-1044,140,889,560,154,20,1,0,-5040,

%U -8028,-64,6363,4809,1638,294,27,1,0,-40320,-69264,-8540,50840,44835,17913,3990,510,35,1

%N Generalized Stirling number triangle of first kind.

%C From _Wolfdieter Lang_, Jun 20 2011: (Start)

%C 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).

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

%C 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)).

%C 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).

%C The row polynomials satisfy also

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

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

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

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

%C The exponential convolution identity is

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

%C n>=0, with symmetry x <-> y.

%C 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.

%C 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).

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

%C 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

%C 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

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

%H M. W. Coffey, M. C. Lettington, <a href="http://arxiv.org/abs/1510.05402">On Fibonacci Polynomial Expressions for Sums of mth Powers, their implications for Faulhaber's Formula and some Theorems of Fermat</a>, arXiv:1510.05402 [math.NT], 2015.

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

%F 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

%F 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

%F From _Wolfdieter Lang_, Jun 20 2011: (Start)

%F 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).

%F 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.

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

%e 1;

%e -1, 1;

%e 0, -1, 1;

%e 0, -1, 0, 1;

%e 0, -2, -1, 2, 1;

%e 0, -6, -5, 5, 5, 1;

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

%e ...

%e Recurrence:

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

%e Row polynomials:

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

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

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

%p 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

%t 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 *);

%t 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 *)

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

%K easy,sign,tabl

%O 0,12

%A _Vladeta Jovovic_, May 17 2004

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Last modified April 17 04:36 EDT 2021. Contains 343059 sequences. (Running on oeis4.)