%I #10 May 24 2018 04:24:16

%S 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 403,489,614,748,929,1125,1391,1676,2055,2475,3012,3613,4379,5233,

%U 6306,7521,9018,10717,12805,15171,18050,21337,25288,29806,35221,41400,48760

%N Total number of parts of multiplicity 9 in all partitions of n.

%H Alois P. Heinz, <a href="/A222709/b222709.txt">Table of n, a(n) for n = 9..1000</a>

%F G.f.: (x^9/(1-x^9)-x^10/(1-x^10))/Product_{j>0}(1-x^j).

%F a(n) ~ exp(Pi*sqrt(2*n/3)) / (180*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=9, 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=9..60);

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

%t Table[a[n], {n, 9, 60}] (* _Jean-François Alcover_, Apr 30 2018, after _Alois P. Heinz_ *)

%Y Column k=9 of A197126.

%K nonn

%O 9,5

%A _Alois P. Heinz_, Feb 28 2013