A105793 -id:A105793 - cp's OEIS frontend

A105794 Inverse of a generalized Stirling number triangle of first kind.

1, -1, 1, 1, -1, 1, -1, 1, 0, 1, 1, -1, 1, 2, 1, -1, 1, 0, 5, 5, 1, 1, -1, 1, 10, 20, 9, 1, -1, 1, 0, 21, 70, 56, 14, 1, 1, -1, 1, 42, 231, 294, 126, 20, 1, -1, 1, 0, 85, 735, 1407, 924, 246, 27, 1, 1, -1, 1, 170, 2290, 6363, 6027, 2400, 435, 35, 1

Offset: 0

Paul Barry, Apr 20 2005

Inverse of number triangle A105793. Row sums are the generalized Bell numbers A000296.
T(n,k)*(-1)^(n-k) gives the inverse Sheffer matrix of A094645. In the umbral notation (cf. Roman, p. 17, quoted in A094645) this is Sheffer for (1-t,-log(1-t)). - Wolfdieter Lang, Jun 20 2011
From Peter Bala, Jul 10 2013: (Start)
For a graph G and a positive integer k, the graphical Stirling number S(G;k) is the number of partitions of the vertex set of G into k nonempty independent sets (an independent set in G is a subset of the vertices, no two elements of which are adjacent). Omitting the first two rows and columns of this triangle produces the triangle of graphical Stirling numbers of cycles on n vertices (interpreting a 2-cycle as a single edge). In comparison, the classical Stirling numbers of the second kind, A008277, are the graphical Stirling numbers of path graphs on n vertices. See Galvin and Than. See also A227341.
This is the triangle of connection constants for expressing the polynomial sequence (x-1)^n in terms of the falling factorial polynomials, that is, (x-1)^n = Sum_{k = 0..n} T(n,k)*x_(k), where x_(k) = x*(x-1)*...*(x-k+1) denotes the falling factorial.
The row polynomials are a particular case of the Actuarial polynomials - see Roman 4.3.4.  (End)

			The triangle starts with
  n=0:  1;
  n=1: -1,  1;
  n=2:  1, -1, 1;
  n=3: -1,  1, 0, 1;
  n=4:  1, -1, 1, 2, 1;
  n=5: -1,  1, 0, 5, 5, 1;
  ... - _Wolfdieter Lang_, Jun 20 2011

S. Roman, The umbral calculus, Pure and Applied Mathematics 111, Academic Press Inc., New York, 1984. Reprinted by Dover in 2005.

Robert Israel, Table of n, a(n) for n = 0..9869
Peter Bala, Notes on A105794
Peter Bala, A 3 parameter family of generalized Stirling numbers
B. Duncan and R. Peele, Bell and Stirling Numbers for Graphs, Journal of Integer Sequences 12 (2009), article 09.7.1.
D. Galvin and D. T. Thanh, Stirling numbers of forests and cycles, Electr. J. Comb. Vol. 20(1): P73 (2013).
Sophie Morier-Genoud, Counting Coxeter's friezes over a finite field via moduli spaces, arXiv:1907.12790 [math.CO], 2019.

Cf. A000296 (row sums), A105793 (matrix inverse), A227341.

Cf. A007318, A008277, A048993.

B:= Matrix(12,12,shape=triangular[lower],(n,k) -> combinat:-stirling1(n-1,k-1)+(n-1)*combinat:-stirling1(n-2,k-1)):
A:= B^(-1):
seq(seq(A[i,j],j=1..i),i=1..12); # Robert Israel, Jan 19 2015
T := (n, k) -> add((-1)^(n - i)*binomial(n, i)*Stirling2(i, k), i=0..n):
seq(seq(T(n, k), k=0..n), n=0..9);  # Peter Luschny, Feb 15 2025

Table[Sum[(-1)^(n - i)*Binomial[n, i] StirlingS2[i, k], {i, 0, n}], {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Oct 14 2019 *)

Term k in row n is given by {(-1)^(k+n) * (Sum_{j=0..k} (-1)^j * binomial(k,j) * (1-j)^n) / k! }; i.e., a finite difference. - Tom Copeland, Jun 05 2006
O.g.f. for row n = n-th finite difference of the Touchard (Bell) polynomials, T(x,j) and so the e.g.f. for these finite differences and therefore the sequence = exp{x*[exp(t)-1]-t} = exp{t*[T(x,.)-1]} umbrally. - Tom Copeland, Jun 05 2006
The e.g.f. A(x,t) = exp(x*(exp(t)-1)-t) satisfies the partial differential equation x*dA/dx - dA/dt = (1-x)*A.
Recurrence relation: T(n+1,k) = T(n,k-1) + (k-1)*T(n,k).
Let f(x) = ((x-1)/x)*exp(x). For n >= 1, the n-th row polynomial R(n,x) = x*exp(-x)*(x*d/dx)^(n-1)(f(x)) and satisfies the recurrence equation R(n+1,x) = (x-1)*R(n,x) + x*R'(n,x). - Peter Bala, Oct 28 2011
Let f(x) = exp(exp(x)-x). Then R(n,exp(x)) = 1/f(x)*(d/dx)^n(f(x)). Similar formulas hold for A008277, A039755, A111577, A143494 and A154537. - Peter Bala, Mar 01 2012
From Peter Bala, Jul 10 2013: (Start)
T(n,k) = Sum_{i = 0..n-1} (-1)^i*Stirling2(n-1-i,k-1), for n >= 1, k >= 1.
The k-th column o.g.f. is (1/(1+x))*x^k/((1-x)*(1-2*x)*...*(1-(k-1)*x)) (the empty product occurring in the denominator when k = 0 and k = 1 is taken equal to 1).
Dobinski-type formula for the row polynomials: R(n,x) = exp(-x)*Sum_{k >= 0} (k-1)^n*x^k/k!.
Sum_{k = 0..n} binomial(n,k)*R(k,x) = n-th Bell polynomial (n-th row polynomial of A048993). (End)
From Peter Bala, Jan 13 2014: (Start)
T(n,k) = Sum_{i = 0..n} (-1)^(n-i)*binomial(n,i)*Stirling2(i,k).
T(n,k) = Sum_{i = 0..n} (-2)^(n-i)*binomial(n,i)*Stirling2(i+1,k+1).
Matrix product P^(-1) * S = P^(-2) * S1, where P = A007318, S = A048993 and S1 = A008277. (End)
From Werner Schulte, Feb 15 2025: (Start)
Conjecture 1: Sum_{k=0..n} (-1)^(n-k) * T(n, k) * (k+m)! / m! = (m+2)^n for m >= 0.
Conjecture 2: (-1)^n - B(n) = Sum_{k=1..n} (-1)^k * T(n, k) * (k-1)! / (k+1) where B(n) = B(n, 0) is n-th Bernoulli number. (End)

A269953 Triangle read by rows: T(n, k) = Sum_{j=0..n} binomial(-j-1, -n-1)*S1(j, k) where S1 are the Stirling cycle numbers A132393.

Original entry on oeis.org

1, -1, 1, 1, -1, 1, -1, 2, 0, 1, 1, 0, 5, 2, 1, -1, 9, 15, 15, 5, 1, 1, 35, 94, 85, 40, 9, 1, -1, 230, 595, 609, 315, 91, 14, 1, 1, 1624, 4458, 4844, 2779, 924, 182, 20, 1, -1, 13209, 37590, 43238, 26817, 9975, 2310, 330, 27, 1

Offset: 0

Peter Luschny, Apr 12 2016

Replacing the Stirling cycle numbers in the definition by the Stirling set numbers leads to A105794.
From Wolfdieter Lang, Jun 19 2017: (Start)
The triangle t(n, k) = (-1)^(n-k)*T(n, k) is the matrix product of P = A007318 (Pascal) and s1 = A048994 (signed Stirling1). This is Sheffer (exp(t), log(1+t)).
The present triangle T is therefore the Sheffer triangle (exp(-t), -log(1-t)). Note that P is Sheffer (exp(t), t) (of the Appell type). (End)
The triangle T(n,k) is a representative of the parametric family of triangles T(m,n,k), whose columns are the coefficients of the standard expansion of the function f(x) = (-log(1-x))^(k)*exp(-m*x)/k! for the case m=1. See A381082. - Igor Victorovich Statsenko, Feb 14 2025

			Triangle starts:
   1;
  -1,  1;
   1, -1,  1;
  -1,  2,  0,  1;
   1,  0,  5,  2,  1;
  -1,  9, 15, 15,  5,  1;
   1, 35, 94, 85, 40,  9,  1.

Peter Luschny, Extensions of the binomial
Eric Weisstein's World of Mathematics, Poisson-Charlier Polynomial

Columns k=0..4 give A033999, A002741, A381064, A381065, A381066.
Cf. A000166 (row sums), A080956 (diag n,n-1).
Cf. A105793, A105794, A132393, A007318, A048994.
Cf. A001339, A046716, A082030, A095000, A137346.
KummerU(-n,1-n-x,z): this sequence (z=-1), A094816 (z=1), |A137346| (z=2), A327997 (z=3).

A269953 := (n,k) -> add(binomial(-j-1,-n-1)*abs(Stirling1(j,k)), j=0..n):
seq(print(seq(A269953(n, k), k=0..n)), n=0..9);
# Alternative:
egf := exp(-t)*(1-t)^(-x): ser := series(egf, t, 12): p := n -> coeff(ser, t, n):
seq(n!*seq(coeff(p(n), x, k), k=0..n), n=0..9); # Peter Luschny, Oct 28 2019

Flatten[Table[Sum[Binomial[-j-1,-n-1] Abs[StirlingS1[j,k]], {j,0,n}], {n,0,9},{k,0,n}]]
(* Or: *)
p [n_] := HypergeometricU[-n, 1 - n - x, -1];
Table[CoefficientList[p[n], x], {n, 0, 9}] (* Peter Luschny, Oct 28 2019 *)

From Wolfdieter Lang, Jun 19 2017: (Start)
E.g.f. of row polynomials R(n, x) = Sum_{k=0..n} T(n,k)*x^k: exp(-t)/(1 - t)^x.
E.g.f. of column k sequence: exp(-x)*(-log(1-x))^k/k!, k >= 0. (End)
From Peter Bala, Oct 26 2019: (Start)
Let R(n, x) = (-1)^n*Sum_{k >= 0} binomial(n,k)*k!* binomial(-x,k) the n-th row polynomial of this triangle.
R(n, x) = c_n(-x;-1), where c_n(x;a) denotes the n-th Poisson Charlier polynomial.
The series representation e = Sum_{k >= 0} 1/k! is the case n = 0 of the more general result e = n!*Sum_{k >= 0} 1/(k!*R(n,k)*R(n,k+1)), n = 0,2,3,4,.... (End)
R(n, x) = KummerU(-n, 1-n-x, -1). - Peter Luschny, Oct 28 2019

A227342 Expansion of (1 - t)*(1 + t)^x.

Original entry on oeis.org

1, -1, 1, 0, -3, 1, 0, 5, -6, 1, 0, -14, 23, -10, 1, 0, 54, -105, 65, -15, 1, 0, -264, 574, -435, 145, -21, 1, 0, 1560, -3682, 3199, -1330, 280, -28, 1, 0, -10800, 27180, -26124, 12649, -3360, 490, -36, 1, 0, 85680, -227196, 236312, -128205, 40089, -7434, 798, -45, 1

Offset: 0

Peter Bala, Jul 11 2013

Cf. A105793. Inverse array is A227343.
The e.g.f. has the form A(t)*exp(x*B(t)), where A(t) = 1 - t and B(t) = log(1 + t). Thus the row polynomials of this triangle form a Sheffer sequence for the pair (2 - exp(t), exp(t) - 1) (see Roman, p. 17).
Let x_(k) := x*(x-1)*...*(x-k+1) denote the k-th falling factorial polynomial. Define a sequence x_[n] of basis polynomials for the polynomial algebra C[x] by setting x_[0] = 1, and setting x_[n] = x_(n-1)*(x - 2*n + 1) for n >= 1. The sequence begins [1, x-1, x*(x-3), x*(x-1)*(x-5), x*(x-1)*(x-2)*(x-7), ...]. Then this is the triangle of connection constants for expressing the basis polynomials x_[n] as a linear combination of the monomial polynomials x^k, that is, x_[n] = Sum_{k = 0..n} T(n,k) x^k. An example is given below.

			Triangle begins:
n\k|  0       1        2       3        4      5      6    7    8   9
===|==================================================================
0  |  1
1  | -1,      1;
2  |  0,     -3,       1;
3  |  0,      5,      -6,      1;
4  |  0,    -14,      23,    -10,       1;
5  |  0,     54,    -105,     65,     -15,     1;
6  |  0,   -264,     574,   -435,     145,   -21,     1;
7  |  0,   1560,   -3682,   3199,   -1330,   280,   -28,   1;
8  |  0, -10800,   27180, -26124,   12649, -3360,   490, -36,   1;
9  |  0,  85680, -227196, 236312, -128205, 40089, -7434, 798, -45,  1;
...
Connection constants. Row 4 = [0, -14 ,23, -10, 1]:
-14*x + 23*x^2 - 10*x^3 + x^4 = x*(x-1)*(x-2)*(x-7) = x_[4].

S. Roman, The umbral calculus, Pure and Applied Mathematics 111, Academic Press Inc., New York, 1984. Reprinted by Dover in 2005.

Eric Weisstein's World of Mathematics, Sheffer Sequence

Cf. A048994, A105793, A227343 (matrix inverse).

T(n,k) = Stirling1(n,k) - n*Stirling1(n-1,k).
E.g.f.: (1 - t)*(1 + t)^x = 1 + (-1 + x)*t + (-3*x + x^2)*t^2/2! + (5*x - 6*x^2 + x^3)*t^3/3! + ....
E.g.f. for column k: (1/k!)*(1 - t)*(log(1 + t))^k.
The row polynomials R(n,x) satisfy the Sheffer identity R(n,x + y) = Sum_{k = 0..n} binomial(n,k)*y_(k)*R(n-k,x), where y_(k) is the falling factorial. As a particular case we have the identity R(n,x + 1) - R(n,x) = n*R(n-1,x) for n >= 1.

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A105794 Inverse of a generalized Stirling number triangle of first kind.

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A269953 Triangle read by rows: T(n, k) = Sum_{j=0..n} binomial(-j-1, -n-1)*S1(j, k) where S1 are the Stirling cycle numbers A132393.

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A227342 Expansion of (1 - t)*(1 + t)^x.

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