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A222710 Total number of parts of multiplicity 10 in all partitions of n.

%I A222710 #10 May 24 2018 04:24:38
%S A222710 1,0,1,1,2,2,4,4,7,8,13,15,22,26,37,45,61,74,99,120,158,192,247,300,
%T A222710 382,463,582,705,877,1059,1309,1575,1931,2319,2823,3381,4094,4886,
%U A222710 5886,7007,8401,9971,11905,14090,16756,19781,23436,27594,32586,38268,45050
%N A222710 Total number of parts of multiplicity 10 in all partitions of n.
%H A222710 Alois P. Heinz, <a href="/A222710/b222710.txt">Table of n, a(n) for n = 10..1000</a>
%F A222710 G.f.: (x^10/(1-x^10)-x^11/(1-x^11))/Product_{j>0}(1-x^j).
%F A222710 a(n) ~ exp(Pi*sqrt(2*n/3)) / (220*Pi*sqrt(2*n)). - _Vaclav Kotesovec_, May 24 2018
%p A222710 b:= proc(n, p) option remember; `if`(n=0, [1, 0], `if`(p<1, [0, 0],
%p A222710       add((l->`if`(m=10, l+[0, l[1]], l))(b(n-p*m, p-1)), m=0..n/p)))
%p A222710     end:
%p A222710 a:= n-> b(n, n)[2]:
%p A222710 seq(a(n), n=10..60);
%t A222710 b[n_, p_] := b[n, p] = If[n == 0, {1, 0}, If[p < 1, {0, 0}, Sum[Function[l, If[m == 10, l + {0, l[[1]]}, l]][b[n - p*m, p - 1]], {m, 0, n/p}]]];
%t A222710 a[n_] := b[n, n][[2]];
%t A222710 Table[a[n], {n, 10, 60}] (* _Jean-François Alcover_, Apr 30 2018, after _Alois P. Heinz_ *)
%Y A222710 Column k=10 of A197126.
%K A222710 nonn
%O A222710 10,5
%A A222710 _Alois P. Heinz_, Feb 28 2013