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

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

0,2

Comments

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.

Links

Table of n, a(n) for n=0..44.

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.

Formula

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.

Example

[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;

Maple

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);

Crossrefs

A361649 (row sums).

Triangle for m=0: A130534.

Triangle for m=1: A376863.

Sequence in context: A027759 A197654 A296043 * A336600 A336599 A066833

Adjacent sequences: A377055 A377056 A377057 * A377059 A377060 A377061

Keyword

nonn

Author

Igor Victorovich Statsenko, Oct 14 2024

Status

approved