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Number of partitions of n into parts of 18 kinds.
2

%I #24 Mar 27 2017 21:32:16

%S 1,18,189,1482,9576,53676,269325,1235286,5256711,20985272,79260723,

%T 285139764,982349361,3255488082,10416507579,32281134120,97154549289,

%U 284625019800,813310723925,2270826800172,6204926551824,16615751700618

%N Number of partitions of n into parts of 18 kinds.

%C a(n) is Euler transform of A010857. - _Alois P. Heinz_, Oct 17 2008

%H Seiichi Manyama, <a href="/A023016/b023016.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Pro#1mxtok">Index entries for expansions of Product_{k >= 1} (1-x^k)^m</a>

%F a(0) = 1, a(n) = (18/n)*Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - _Seiichi Manyama_, Mar 27 2017

%p with(numtheory): a:= proc(n) option remember; `if`(n=0, 1, add(add(d*18, d=divisors(j)) *a(n-j), j=1..n)/n) end: seq(a(n), n=0..40); # _Alois P. Heinz_, Oct 17 2008

%t CoefficientList[Series[1/QPochhammer[x]^18, {x, 0, 30}], x] (* _Indranil Ghosh_, Mar 27 2017 *)

%o (PARI) Vec(1/eta(x)^18 + O(x^30)) \\ _Indranil Ghosh_, Mar 27 2017

%Y Cf. 18th column of A144064. - _Alois P. Heinz_, Oct 17 2008

%K nonn

%O 0,2

%A _David W. Wilson_