A222706 - OEIS

%I #10 May 24 2018 04:23:17

%S 1,0,1,1,2,2,5,5,8,10,15,18,28,33,47,58,79,97,132,161,212,262,337,414,

%T 531,648,818,1001,1249,1519,1887,2285,2812,3401,4155,5004,6086,7301,

%U 8827,10565,12708,15155,18162,21587,25757,30539,36296,42904,50832,59915

%N Total number of parts of multiplicity 6 in all partitions of n.

%H Alois P. Heinz, <a href="/A222706/b222706.txt">Table of n, a(n) for n = 6..1000</a>

%F G.f.: (x^6/(1-x^6)-x^7/(1-x^7))/Product_{j>0}(1-x^j).

%F a(n) ~ exp(Pi*sqrt(2*n/3)) / (84*Pi*sqrt(2*n)). - _Vaclav Kotesovec_, May 24 2018

%p b:= proc(n, p) option remember; `if`(n=0, [1, 0], `if`(p<1, [0, 0],

%p       add((l->`if`(m=6, l+[0, l[1]], l))(b(n-p*m, p-1)), m=0..n/p)))

%p     end:

%p a:= n-> b(n, n)[2]:

%p seq(a(n), n=6..60);

%t b[n_, p_] := b[n, p] = If[n == 0, {1, 0}, If[p < 1, {0, 0}, Sum[Function[l, If[m == 6, l + {0, l[[1]]}, l]][b[n - p*m, p - 1]], {m, 0, n/p}]]];

%t a[n_] := b[n, n][[2]];

%t Table[a[n], {n, 6, 60}] (* _Jean-François Alcover_, Apr 30 2018, after _Alois P. Heinz_ *)

%Y Column k=6 of A197126.

%K nonn

%O 6,5

%A _Alois P. Heinz_, Feb 28 2013