The OEIS is supported by the many generous donors to the OEIS Foundation.
%I #7 Mar 11 2018 20:26:28
%S 1,3,38,786,22888,857800,39316464,2130380560,133222474368,
%T 9443111340672,748168002970880,65520799156209408,6284786657494483968,
%U 655287035001111884800,73792143714173551392768,8925528145554323771934720,1154065253662722209679572992,158849709577131169400652988416
%N a(n) = n! * [x^n] (Sum_{k=0..n} prime(k+1)*x^k/k!)^n.
%H Alois P. Heinz, <a href="/A300631/b300631.txt">Table of n, a(n) for n = 0..333</a>
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%e The table of coefficients of x^k in expansion of e.g.f. (Sum_{k>=0} prime(k+1)*x^k/k!)^n begins:
%e n = 0: (1), 0, 0, 0, 0, 0, ... (A000007)
%e n = 1: 2, (3), 5, 7, 11, 13, ... (A000040, with offset 0)
%e n = 2: 4, 12, (38), 118, 362, 1082, ... (A014345)
%e n = 3: 8, 36, 168, (786), 3660, 16866, ... (A014347)
%e n = 4: 16, 96, 592, 3680, (22888), 141776, ... (A014352)
%e n = 5: 32, 240, 1840, 14240, 110560, (857800), ...
%p b:= proc(n, k) option remember; `if`(k=1, ithprime(n+1), add(
%p b(j, floor(k/2))*b(n-j, ceil(k/2))*binomial(n, j), j=0..n))
%p end:
%p a:= n-> `if`(n=0, 1, b(n$2)):
%p seq(a(n), n=0..20); # _Alois P. Heinz_, Mar 10 2018
%t Table[n! SeriesCoefficient[Sum[Prime[k + 1] x^k/k!, {k, 0, n}]^n, {x, 0, n}], {n, 0, 17}]
%Y Cf. A000007, A000040, A014345, A014347, A014352.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Mar 10 2018