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

0,2

Links

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

Formula

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

Example

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, ...

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);

Crossrefs

Row sums are A325140.

Columns are: A000142, A001711, A001717, A001723.

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

Sequence in context: A345323 A052636 A172430 * A084312 A066752 A192329

Adjacent sequences: A325136 A325137 A325138 * A325140 A325141 A325142

Keyword

nonn,tabl

Author

Peter Luschny, Apr 15 2019

Status

approved