A325139 Triangle T(n, k) = [t^n] Gamma(n + k + m + t)/Gamma(k + m + t) for m = 2 and 0 <= k <= n, read by rows.

1, 2, 1, 6, 7, 1, 24, 47, 15, 1, 120, 342, 179, 26, 1, 720, 2754, 2070, 485, 40, 1, 5040, 24552, 24574, 8175, 1075, 57, 1, 40320, 241128, 305956, 134449, 24885, 2086, 77, 1, 362880, 2592720, 4028156, 2231012, 541849, 63504, 3682, 100, 1

Offset: 0

Author

Peter Luschny, Apr 15 2019

Keywords

• nonn
• tabl

Examples

			0:        1;
1:        2,        1;
2:        6,        7,        1;
3:       24,       47,       15,        1;
4:      120,      342,      179,       26,        1;
5:      720,     2754,     2070,      485,       40,       1;
6:     5040,    24552,    24574,     8175,     1075,      57,      1;
7:    40320,   241128,   305956,   134449,    24885,    2086,     77,    1;
8:   362880,  2592720,  4028156,  2231012,   541849,   63504,   3682,  100,   1;
9:  3628800, 30334320, 56231712, 37972304, 11563650, 1768809, 142632, 6054, 126, 1;
A:  A000142,  A001711,  A001717,  A001723, ...
		

Crossrefs

Row sums are A325140.

Columns are: A000142, A001711, A001717, A001723.

Family: A307419 (m=0), A325137 (m=1), this sequence (m=2).

Programs

Maple

T := (n, k) -> add(binomial(j+k, k)*(k+2)^j*abs(Stirling1(n, j+k)), j=0..n-k):
seq(seq(T(n, k), k=0..n), n=0..8);
# Note that for n > 16 Maple fails (at least in some versions) to compute the
# terms properly. Inserting 'simplify' or numerical evaluation might help.
A325139Row := proc(n) local ogf, ser; ogf := (n, k) -> GAMMA(n+k+2+x)/GAMMA(k+2+x);
ser := (n, k) -> series(ogf(n,k), x, k+2); seq(coeff(ser(n,k), x, k), k=0..n) end:
seq(A325139Row(n), n=0..9);

Formula

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