A300631 a(n) = n! * [x^n] (Sum_{k=0..n} prime(k+1)*x^k/k!)^n.

1, 3, 38, 786, 22888, 857800, 39316464, 2130380560, 133222474368, 9443111340672, 748168002970880, 65520799156209408, 6284786657494483968, 655287035001111884800, 73792143714173551392768, 8925528145554323771934720, 1154065253662722209679572992, 158849709577131169400652988416

Offset: 0

Author

Ilya Gutkovskiy, Mar 10 2018

Keywords

• nonn

Examples

			The table of coefficients of x^k in expansion of e.g.f. (Sum_{k>=0} prime(k+1)*x^k/k!)^n begins:
n = 0:  (1),   0,     0,      0,       0,        0,  ... (A000007)
n = 1:   2,   (3),    5,      7,      11,       13,  ... (A000040, with offset 0)
n = 2:   4,   12,   (38),   118,     362,     1082,  ... (A014345)
n = 3:   8,   36,   168,   (786),   3660,    16866,  ... (A014347)
n = 4:  16,   96,   592,   3680,  (22888),  141776,  ... (A014352)
n = 5:  32,  240,  1840,  14240,  110560,  (857800), ...
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..333
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A000007, A000040, A014345, A014347, A014352.

Programs

Maple

b:= proc(n, k) option remember; `if`(k=1, ithprime(n+1), add(
      b(j, floor(k/2))*b(n-j, ceil(k/2))*binomial(n, j), j=0..n))
    end:
a:= n-> `if`(n=0, 1, b(n$2)):
seq(a(n), n=0..20);  # Alois P. Heinz, Mar 10 2018

Mathematica

Table[n! SeriesCoefficient[Sum[Prime[k + 1] x^k/k!, {k, 0, n}]^n, {x, 0, n}], {n, 0, 17}]