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

0,4

Comments

Sister triangle of A307419.

Links

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

Formula

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

Example

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, ...

Maple

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);

Crossrefs

Row sums: A325138.

Columns are: A000142, A001705, A001712, A001718, A001724.

Cf. A307419.

Sequence in context: A084245 A174232 A065224 * A321966 A304822 A165278

Adjacent sequences: A325134 A325135 A325136 * A325138 A325139 A325140

Keyword

nonn,tabl

Author

Peter Luschny, Apr 13 2019

Status

approved