A376863 - OEIS

A376863

Triangle of generalized Stirling numbers of the lower level of the hierarchy (section m=1).

1, 3, 1, 13, 7, 1, 73, 50, 12, 1, 501, 400, 125, 18, 1, 4051, 3609, 1335, 255, 25, 1, 37633, 36463, 15214, 3485, 460, 33, 1, 394353, 408694, 186949, 48769, 7805, 763, 42, 1, 4596553, 5036792, 2479602, 714364, 131299, 15708, 1190, 52, 1, 58941091, 67714809, 35419350, 11045558, 2256933, 312375, 29190, 1770, 63, 1, 824073141, 986271823, 543025851, 180766890, 40194965, 6221397, 676893, 50970, 2535, 75, 1

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..65.

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. 19-12. 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 = 1.

EXAMPLE

Triangle starts:

[0] 1;

[1] 3, 1;

[2] 13, 7, 1;

[3] 73, 50, 12, 1;

[4] 501, 400, 125, 18, 1;

[5] 4051, 3609, 1335, 255, 25, 1;

[6] 37633, 36463, 15214, 3485, 460, 33, 1;

[7] 394353, 408694, 186949, 48769, 7805, 763, 42, 1;

[8] 4596553, 5036792, 2479602, 714364, 131299, 15708, 1190, 52, 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:=1:seq(seq(T(m, n, k), k=0..n), n=0..10);

CROSSREFS

A000262 (column 0), A052852 (row sums).

Triangle for m=0: A130534.

Sequence in context: A118384 A341725 A258239 * A133176 A089435 A152474

Adjacent sequences: A376859 A376860 A376862 * A376864 A376865 A376866

KEYWORD

nonn,tabl

AUTHOR

Igor Victorovich Statsenko, Oct 07 2024

STATUS

approved