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A222737 Total sum of parts of multiplicity 9 in all partitions of n.

%I A222737 #10 May 29 2018 09:19:12
%S A222737 1,0,1,1,2,2,4,4,7,10,14,16,25,30,42,53,71,88,121,148,195,241,312,384,
%T A222737 494,605,765,943,1179,1441,1796,2181,2694,3273,4011,4849,5922,7130,
%U A222737 8652,10398,12552,15021,18072,21558,25816,30729,36649,43480,51705,61163
%N A222737 Total sum of parts of multiplicity 9 in all partitions of n.
%H A222737 Alois P. Heinz, <a href="/A222737/b222737.txt">Table of n, a(n) for n = 9..1000</a>
%F A222737 G.f.: (x^9/(1-x^9)^2-x^10/(1-x^10)^2)/Product_{i>=1}(1-x^i).
%F A222737 a(n) ~ 19 * sqrt(3) * exp(Pi*sqrt(2*n/3)) / (16200 * Pi^2). - _Vaclav Kotesovec_, May 29 2018
%p A222737 b:= proc(n, p) option remember; `if`(n=0, [1, 0], `if`(p<1, [0, 0],
%p A222737       add((l->`if`(m=9, l+[0, l[1]*p], l))(b(n-p*m, p-1)), m=0..n/p)))
%p A222737     end:
%p A222737 a:= n-> b(n, n)[2]:
%p A222737 seq(a(n), n=9..60);
%t A222737 b[n_, p_] := b[n, p] = If[n == 0 && p == 0, {1, 0}, If[p == 0, Array[0&, n+2], Sum[Function[l, ReplacePart[l, m+2 -> p*l[[1]] + l[[m+2]]]][Join[b[n-p*m, p-1], Array[0&, p*m]]], {m, 0, n/p}]]]; a[n_] := b[n, n][[11]]; Table[a[n], {n, 9, 60}] (* _Jean-François Alcover_, Jan 24 2014, after _Alois P. Heinz_ *)
%Y A222737 Column k=9 of A222730.
%K A222737 nonn
%O A222737 9,5
%A A222737 _Alois P. Heinz_, Mar 03 2013