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Total number of parts of multiplicity 8 in all partitions of n.
2

%I #10 May 24 2018 04:23:58

%S 1,0,1,1,2,2,4,4,8,9,13,16,24,28,40,49,67,82,110,133,176,215,276,338,

%T 432,524,661,804,1001,1214,1502,1812,2228,2683,3270,3928,4766,5699,

%U 6879,8208,9855,11724,14020,16624,19804,23423,27788,32781,38770,45603,53762

%N Total number of parts of multiplicity 8 in all partitions of n.

%H Alois P. Heinz, <a href="/A222708/b222708.txt">Table of n, a(n) for n = 8..1000</a>

%F G.f.: (x^8/(1-x^8)-x^9/(1-x^9))/Product_{j>0}(1-x^j).

%F a(n) ~ exp(Pi*sqrt(2*n/3)) / (144*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=8, 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=8..60);

%t b[n_, p_] := b[n, p] = If[n == 0, {1, 0}, If[p < 1, {0, 0}, Sum[Function[l, If[m == 8, 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, 8, 60}] (* _Jean-François Alcover_, Apr 30 2018, after _Alois P. Heinz_ *)

%Y Column k=8 of A197126.

%K nonn

%O 8,5

%A _Alois P. Heinz_, Feb 28 2013