A370518 -id:A370518 - cp's OEIS frontend

A094645 Triangle of generalized Stirling numbers of the first kind.

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

Offset: 0

Vladeta Jovovic, May 17 2004

From Wolfdieter Lang, Jun 20 2011: (Start)
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).
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 also satisfy
  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 at 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_{k=0..n} binomial(n,k)*s(k,y)*s1(n-k,x),
  n >= 0, with symmetry x <-> y.
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.
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

			Triangle begins
   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).

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

M. W. Coffey and 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.
Igor Victorovich Statsenko, On the ordinal numbers of triangles of generalized special numbers, Innovation science No 2-2, State Ufa, Aeterna Publishing House, 2024, pp. 15-19. In Russian.

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

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[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 *)

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

A377058 Triangle of generalized Stirling numbers of the lower level of the hierarchy (case m=2).

Original entry on oeis.org

1, 5, 1, 32, 11, 1, 248, 113, 18, 1, 2248, 1230, 263, 26, 1, 23272, 14534, 3765, 505, 35, 1, 270400, 186992, 55654, 9115, 865, 45, 1, 3479744, 2612000, 865186, 163779, 19110, 1372, 56, 1, 49079936, 39434448, 14235388, 3013164, 408569, 36288, 2058, 68, 1

Offset: 0

Igor Victorovich Statsenko, Oct 14 2024

nonn

These numbers are a subset of the generalized Stirling numbers introduced in A370518. Therefore, we assume them to be numbers of the lower level of hierarchy with respect to A370518.

			[0]           1;
[1]           5,          1;
[2]          32,         11,         1;
[3]         248,        113,        18,        1;
[4]        2248,       1230,       263,       26,       1;
[5]       23272,      14534,      3765,      505,      35,      1;
[6]      270400,     186992,     55654,     9115,     865,     45,     1;
[7]     3479744,    2612000,    865186,   163779,   19110,   1372,    56,    1;
[8]    49079936,   39434448,  14235388,  3013164,  408569,  36288,  2058,   68,    1;

Igor Victorovich Statsenko, Relationships of "P"-generalized Stirling numbers of the first kind with other generalized Stirling numbers, Innovation science No 10-1, State Ufa, Aeterna Publishing House, 2024, pp. 19-22. In Russian.

A361649 (row sums).
Triangle for m=0: A130534.
Triangle for m=1: A376863.

T := (m,n,k) -> add(add(Stirling1(n-j,k)*binomial(n+m,i)*binomial(n,j)*binomial(j,i)*i!*m^(j-i), j=i..n), i=0..n): m:=2: seq(seq(T(m,n,k), k=0..n), n=0..10);

T(m, n, k) = Sum_{i=0..n} Sum_{j=i..n} Stirling1(n-j, k)*binomial(n+m, i)*binomial(n, j)* binomial(j, i)*i!*m^(j-i), for m = 2.

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A094645 Triangle of generalized Stirling numbers of the first kind.

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