A137597 Triangle read by rows: A008277 * A007318.

1, 2, 1, 5, 5, 1, 15, 22, 9, 1, 52, 99, 61, 14, 1, 203, 471, 385, 135, 20, 1, 877, 2386, 2416, 1140, 260, 27, 1, 4140, 12867, 15470, 9156, 2835, 455, 35, 1, 21147, 73681, 102215, 72590, 28441, 6230, 742, 44, 1

Offset

1,2

Comments

Row sums = A035009 starting (1, 3, 11, 47, 227, ...).

Links

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows n = 1..150, flattened)

Zhanar Berikkyzy, Pamela E. Harris, Anna Pun, Catherine Yan, and Chenchen Zhao, Combinatorial Identities for Vacillating Tableaux, arXiv:2308.14183 [math.CO], 2023. See p. 16.

Formula

A008277 * A007318 as infinite lower triangular matrices.

T(n,k) = Sum_{j=k..n} C(j-1,k-1)*Stirling2(n,j).

From Natalia L. Skirrow, Jun 26 2026: (Start)

D-finite with T(n+1,k) = k*T(n,k+1) + (k+1)*T(n,k) + T(n,k-1).

T(n,k) = Sum_{j=k..n} C(n,j) * Bell(n-j) * Sum_{i=k..j} (-1)^(i-k)*Stirling2(j,i), where Bell(n) = A000110(n).

T(n,k) = -Sum_{j=0..n} C(n,j) * Bell(n-j) * Sum_{i=0..k-1} (-1)^(k-i)*Stirling2(j,i).

E.g.f.: (e^((1+y)*(e^x-1)) - 1) * y/(1+y).

Compare (e^((e^x-1)/(1-y)) - 1) * y/(1-y), e.g.f. of this triangle's binomial transform w.r.t. k.

k-th column e.g.f.:

(-1)^k* e^(e^x-1) * Sum_{i>=k} (1-e^x)^i/i!;

= (-1)^k*(1 - e^(e^x-1) * Sum_{i=0..k-1} (1-e^x)^i/i!);

= (-1)^k*gamma(k,1-e^x)/(k-1)!, where gamma is the lower incomplete gamma function.

For x <= 0, k-th column e.g.f. is (-1)^k times the probability that a Poisson random variable with expectation 1-e^x is >=k.

O.g.f.: hypergeom([1],[1-1/x],-1-y) * y/(y+1).

Column o.g.f.: hypergeom([k],[1-1/x+k],-1) / (1/x-1)_k, where x_k = x!/(x-k)! is a falling power.

T(n,k) = Sum_{l=0..k-1} (-1)^(k-1-l) * E(k-1-l) * A108087(n,l)/l! where E(n) = Sum_{j=0..n} 1/j! = A120265(n)/A061355(n).

(End)

Example

First few rows of the triangle:
    1;
    2,   1;
    5,   5,   1;
   15,  22,   9,   1;
   52,  99,  61,  14,  1;
  203, 471, 385, 135, 20, 1;
  ...

Maple

T:= (n, k)-> add(Stirling2(n, j)*binomial(j-1, k-1), j=k..n):
seq(seq(T(n, k), k=1..n), n=1..10);  # Alois P. Heinz, Sep 03 2019

Mathematica

Table[Sum[StirlingS2[n, j]*Binomial[j - 1, k - 1], {j, k, n}], {n, 9}, {k, n}] // Flatten (* Michael De Vlieger, Aug 31 2023 *)

Prog

(Python)
from math import comb
from sympy.functions.combinatorial.numbers import stirling
A137597=lambda n, k: sum(comb(j-1, k-1)*stirling(n, j, kind=2) for j in range(k, n+1)) # Natalia L. Skirrow, Jun 26 2026

Crossrefs

Cf. A000110, A108087, A120265/A061355.

Cf. A035009, A008277, A007318.

Cf. A126350.

Sequence in context: A107842 A126216 A124733 * A059340 A248727 A270250

Adjacent sequences: A137594 A137595 A137596 * A137598 A137599 A137600

Keyword

nonn,tabl,changed

Author

Gary W. Adamson, Jan 29 2008

Status

approved