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A222709 Total number of parts of multiplicity 9 in all partitions of n.

%I A222709 #10 May 24 2018 04:24:16
%S A222709 1,0,1,1,2,2,4,4,7,9,13,15,23,27,38,47,63,77,104,126,165,202,259,316,
%T A222709 403,489,614,748,929,1125,1391,1676,2055,2475,3012,3613,4379,5233,
%U A222709 6306,7521,9018,10717,12805,15171,18050,21337,25288,29806,35221,41400,48760
%N A222709 Total number of parts of multiplicity 9 in all partitions of n.
%H A222709 Alois P. Heinz, <a href="/A222709/b222709.txt">Table of n, a(n) for n = 9..1000</a>
%F A222709 G.f.: (x^9/(1-x^9)-x^10/(1-x^10))/Product_{j>0}(1-x^j).
%F A222709 a(n) ~ exp(Pi*sqrt(2*n/3)) / (180*Pi*sqrt(2*n)). - _Vaclav Kotesovec_, May 24 2018
%p A222709 b:= proc(n, p) option remember; `if`(n=0, [1, 0], `if`(p<1, [0, 0],
%p A222709       add((l->`if`(m=9, l+[0, l[1]], l))(b(n-p*m, p-1)), m=0..n/p)))
%p A222709     end:
%p A222709 a:= n-> b(n, n)[2]:
%p A222709 seq(a(n), n=9..60);
%t A222709 b[n_, p_] := b[n, p] = If[n == 0, {1, 0}, If[p < 1, {0, 0}, Sum[Function[l, If[m == 9, l + {0, l[[1]]}, l]][b[n - p*m, p - 1]], {m, 0, n/p}]]];
%t A222709 a[n_] := b[n, n][[2]];
%t A222709 Table[a[n], {n, 9, 60}] (* _Jean-François Alcover_, Apr 30 2018, after _Alois P. Heinz_ *)
%Y A222709 Column k=9 of A197126.
%K A222709 nonn
%O A222709 9,5
%A A222709 _Alois P. Heinz_, Feb 28 2013