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Number of partitions of n into parts of 16 kinds.
3

%I #21 Mar 27 2017 21:32:01

%S 1,16,152,1088,6460,33440,155584,663936,2636326,9845040,34861152,

%T 117809728,381946360,1193074144,3603543040,10556065152,30068145905,

%U 83466484112,226236086512,599785472000,1557643542308,3967888347232,9926348625408,24413219138816

%N Number of partitions of n into parts of 16 kinds.

%C a(n) is Euler transform of A010855. - _Alois P. Heinz_, Oct 17 2008

%H Alois P. Heinz, <a href="/A023014/b023014.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>

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%F G.f.: Product_{m>=1} 1/(1-x^m)^16.

%F a(0) = 1, a(n) = (16/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*16, 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]^16, {x, 0, 30}], x] (* _Indranil Ghosh_, Mar 27 2017 *)

%o (PARI) Vec(1/eta(x)^16 + O(x^30)) \\ _Indranil Ghosh_, Mar 27 2017

%Y Cf. 16th column of A144064. - _Alois P. Heinz_, Oct 17 2008

%K nonn

%O 0,2

%A _David W. Wilson_