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A222735 Total sum of parts of multiplicity 7 in all partitions of n.

%I A222735 #10 May 29 2018 09:19:24
%S A222735 1,0,1,1,2,2,4,6,9,10,16,20,29,36,53,66,91,112,152,190,251,315,409,
%T A222735 510,655,809,1029,1271,1602,1967,2457,3009,3729,4543,5595,6801,8321,
%U A222735 10069,12258,14783,17906,21511,25947,31073,37315,44542,53285,63415,75587,89687
%N A222735 Total sum of parts of multiplicity 7 in all partitions of n.
%H A222735 Alois P. Heinz, <a href="/A222735/b222735.txt">Table of n, a(n) for n = 7..1000</a>
%F A222735 G.f.: (x^7/(1-x^7)^2-x^8/(1-x^8)^2)/Product_{i>=1}(1-x^i).
%F A222735 a(n) ~ 15 * sqrt(3) * exp(Pi*sqrt(2*n/3)) / (6272 * Pi^2). - _Vaclav Kotesovec_, May 29 2018
%p A222735 b:= proc(n, p) option remember; `if`(n=0, [1, 0], `if`(p<1, [0, 0],
%p A222735       add((l->`if`(m=7, l+[0, l[1]*p], l))(b(n-p*m, p-1)), m=0..n/p)))
%p A222735     end:
%p A222735 a:= n-> b(n, n)[2]:
%p A222735 seq(a(n), n=7..60);
%t A222735 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][[9]]; Table[a[n], {n, 7, 60}] (* _Jean-François Alcover_, Jan 24 2014, after _Alois P. Heinz_ *)
%Y A222735 Column k=7 of A222730.
%K A222735 nonn
%O A222735 7,5
%A A222735 _Alois P. Heinz_, Mar 03 2013