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A300631
a(n) = n! * [x^n] (Sum_{k=0..n} prime(k+1)*x^k/k!)^n.
1
1, 3, 38, 786, 22888, 857800, 39316464, 2130380560, 133222474368, 9443111340672, 748168002970880, 65520799156209408, 6284786657494483968, 655287035001111884800, 73792143714173551392768, 8925528145554323771934720, 1154065253662722209679572992, 158849709577131169400652988416
OFFSET
0,2
LINKS
N. J. A. Sloane, Transforms
EXAMPLE
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), ...
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}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Mar 10 2018
STATUS
approved