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A325137 Triangle T(n, k) = [x^n] (n + k + x)!/(k + x)! for 0 <= k <= n, read by rows. 2
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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified April 9 05:24 EDT 2020. Contains 333343 sequences. (Running on oeis4.)