A307419 -id:A307419 - cp's OEIS frontend

A325137 Triangle T(n, k) = [x^n] (n + k + x)!/(k + x)! for 0 <= k <= n, read by rows.

1, 1, 1, 2, 5, 1, 6, 26, 12, 1, 24, 154, 119, 22, 1, 120, 1044, 1175, 355, 35, 1, 720, 8028, 12154, 5265, 835, 51, 1, 5040, 69264, 133938, 77224, 17360, 1687, 70, 1, 40320, 663696, 1580508, 1155420, 342769, 46816, 3066, 92, 1

Offset: 0

Peter Luschny, Apr 13 2019

Sister triangle of A307419.

			Triangle starts:
[0]      1
[1]      1,       1
[2]      2,       5,        1
[3]      6,      26,       12,        1
[4]     24,     154,      119,       22,       1
[5]    120,    1044,     1175,      355,      35,       1
[6]    720,    8028,    12154,     5265,     835,      51,      1
[7]   5040,   69264,   133938,    77224,   17360,    1687,     70,    1
[8]  40320,  663696,  1580508,  1155420,  342769,   46816,   3066,   92,   1
[9] 362880, 6999840, 19978308, 17893196, 6687009, 1197273, 109494, 5154, 117, 1
   A000142, A001705,  A001712,  A001718, A001724, ...

Row sums: A325138.
Columns are: A000142, A001705, A001712, A001718, A001724.
Cf. A307419.

T := (n, k) -> add(binomial(j+k, k)*(k+1)^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.
A325137Row := proc(n) local ogf, ser; ogf := (n, k) -> (n+k+x)!/(k+x)!;
ser := (n, k) -> series(ogf(n,k),x,k+2); seq(coeff(ser(n,k),x,k), k=0..n) end: seq(A325137Row(n), n=0..8);

T(n, k) = Sum_{j=0..n-k} binomial(j+k, k)*|Stirling1(n, j+k)|*(k+1)^j.

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.

Original entry on oeis.org

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

Peter Luschny, Apr 15 2019

			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, ...

Row sums are A325140.
Columns are: A000142,  A001711,  A001717,  A001723.
Family: A307419 (m=0), A325137 (m=1), this sequence (m=2).

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);

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

cp's OEIS Frontend

A325137 Triangle T(n, k) = [x^n] (n + k + x)!/(k + x)! for 0 <= k <= n, read by rows.

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