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

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