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A299168 Number of ordered ways of writing n-th prime number as a sum of n primes.

%I A299168 #9 Feb 13 2021 09:12:14
%S A299168 1,0,0,0,5,6,42,64,387,5480,10461,113256,507390,1071084,4882635,
%T A299168 44984560,382362589,891350154,7469477771,33066211100,78673599501,
%U A299168 649785780710,2884039365010,22986956007816,306912836483025,1361558306986280,3519406658042964
%N A299168 Number of ordered ways of writing n-th prime number as a sum of n primes.
%H A299168 Alois P. Heinz, <a href="/A299168/b299168.txt">Table of n, a(n) for n = 1..400</a>
%F A299168 a(n) = [x^prime(n)] (Sum_{k>=1} x^prime(k))^n.
%e A299168 a(5) = 5 because fifth prime number is 11 and we have [3, 2, 2, 2, 2], [2, 3, 2, 2, 2], [2, 2, 3, 2, 2], [2, 2, 2, 3, 2] and [2, 2, 2, 2, 3].
%p A299168 b:= proc(n, t) option remember;
%p A299168       `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add(
%p A299168       `if`(isprime(j), b(n-j, t-1), 0), j=1..n)))
%p A299168     end:
%p A299168 a:= n-> b(ithprime(n), n):
%p A299168 seq(a(n), n=1..30);  # _Alois P. Heinz_, Feb 13 2021
%t A299168 Table[SeriesCoefficient[Sum[x^Prime[k], {k, 1, n}]^n, {x, 0, Prime[n]}], {n, 1, 27}]
%Y A299168 Cf. A000040, A000586, A000607, A023360, A056768, A070215, A073610, A098238, A117278, A121303, A219180, A224344, A259254, A265112, A341459.
%K A299168 nonn
%O A299168 1,5
%A A299168 _Ilya Gutkovskiy_, Feb 04 2018