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A094645 Triangle of generalized Stirling numbers of the first kind.

%I A094645 #59 Mar 24 2024 13:07:27
%S A094645 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 A094645 0,-120,-154,49,140,70,14,1,0,-720,-1044,140,889,560,154,20,1,0,-5040,
%U A094645 -8028,-64,6363,4809,1638,294,27,1,0,-40320,-69264,-8540,50840,44835,17913,3990,510,35,1
%N A094645 Triangle of generalized Stirling numbers of the first kind.
%C A094645 From _Wolfdieter Lang_, Jun 20 2011: (Start)
%C A094645 The row polynomials s(n,x) := Sum_{j=0..n} T(n,k)*x^k satisfy risefac(x-1,n) = s(n,x), with the rising factorials risefac(x-1,n) := Product_{j=0..n-1} (x-1+j), 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 A094645 This is the lower triangular Sheffer array with e.g.f.
%C A094645   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 A094645 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 A094645 The row polynomials also satisfy
%C A094645   s(n,x) - s(n,x-1) = n*s(n-1,x), n > 1, s(0,x) = 1
%C A094645   (from the Meixner identity, see the Meixner reference given at A060338).
%C A094645 The row polynomials satisfy as well (from corollary 3.7.2. p. 50 of the Roman reference)
%C A094645   s(n,x) = (x-1)*s(n-1,x+1), n >= 1, s(0,n) = 1.
%C A094645 The exponential convolution identity is
%C A094645   s(n,x+y) = Sum_{k=0..n} binomial(n,k)*s(k,y)*s1(n-k,x),
%C A094645   n >= 0, with symmetry x <-> y.
%C A094645 The row sums are 1 for n=0 and 0 otherwise, and the alternating row sums are 1,-2,2, followed by zeros, with e.g.f. (1-x)^2.
%C A094645 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 A094645 The inverse Sheffer matrix is ((-1)^(n-k))*A105794(n,k) with e.g.f. exp(z)*exp(x*(1-exp(-z))). (End)
%C A094645 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 A094645 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 A094645 S. Roman, The Umbral Calculus, Academic Press, New York, 1984.
%H A094645 M. W. Coffey and 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.
%H A094645 Igor Victorovich Statsenko, <a href="https://aeterna-ufa.ru/sbornik/IN-2024-02-2.pdf#page=15">On the ordinal numbers of triangles of generalized special numbers</a>, Innovation science No 2-2, State Ufa, Aeterna Publishing House, 2024, pp. 15-19. In Russian.
%F A094645 E.g.f.: (1-y)^(1-x).
%F A094645 Sum_{k=0..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 A094645 If we define f(n,i,a) = Sum_{k=0..n-i} binomial(n,k)*Stirling1(n-k,i)*Product_{j=0..k-1} (-a-j), then |T(n,i)| = |f(n,i,-1)|, for n=1,2,...; i=0..n. - _Milan Janjic_, Dec 21 2008
%F A094645 From _Wolfdieter Lang_, Jun 20 2011: (Start)
%F A094645 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 A094645 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 A094645 E.g.f. column k: (1-x)*((-log(1-x))^k)/k!. (End)
%F A094645 T(n,k) = Sum_{i=0..n} binomial(n,i)*(n-i)!*Stirling1(i,k)*TC(m,n,i) where TC(m,n,k) = Sum_{i=0..n-k} binomial(n+1,n-k-i)*Stirling2(i+m+1,i+1)*(-1)^i, m = 1 for n >= 0. See A130534, A370518 for m=0 and m=2. - _Igor Victorovich Statsenko_, Feb 27 2024
%e A094645 Triangle begins
%e A094645    1;
%e A094645   -1,   1;
%e A094645    0,  -1,   1;
%e A094645    0,  -1,   0,   1;
%e A094645    0,  -2,  -1,   2,   1;
%e A094645    0,  -6,  -5,   5,   5,   1;
%e A094645    0, -24, -26,  15,  25,   9,   1;
%e A094645    ...
%e A094645 Recurrence:
%e A094645   -2 = T(4,1) = T(3,0) + (4-2)*T(3,1) = 0 + 2*(-1).
%e A094645 Row polynomials:
%e A094645   s(3,x) = -x+x^3 = (x-1)*s1(2,x) = (x-1)*(x+x^2).
%e A094645   s(3,x) = (x-1)*s(2,x+1) = (x-1)*(-(x+1)+(x+1)^2).
%e A094645   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 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 A094645 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 A094645 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 A094645 Cf. A049444, A049458, A094646, A132393, A105794.
%K A094645 easy,sign,tabl
%O A094645 0,12
%A A094645 _Vladeta Jovovic_, May 17 2004