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A212545 Number of partitions of n containing at least one part m-5 if m is the largest part.

%I A212545 #11 Dec 07 2020 09:43:16
%S A212545 0,0,1,1,3,4,8,11,19,25,39,52,75,98,137,175,236,300,393,493,635,787,
%T A212545 997,1227,1531,1869,2309,2796,3420,4119,4994,5979,7201,8574,10260,
%U A212545 12164,14470,17082,20225,23778,28025,32838,38542,45011,52642,61286,71434,82937
%N A212545 Number of partitions of n containing at least one part m-5 if m is the largest part.
%H A212545 Alois P. Heinz, <a href="/A212545/b212545.txt">Table of n, a(n) for n = 5..1000</a>
%F A212545 G.f.: Sum_{i>0} x^(2*i+5) / Product_{j=1..5+i} (1-x^j).
%e A212545 a(7) = 1: [6,1].
%e A212545 a(8) = 1: [6,1,1].
%e A212545 a(9) = 3: [6,1,1,1], [6,2,1], [7,2].
%e A212545 a(10) = 4: [6,1,1,1,1], [6,2,1,1], [6,3,1], [7,2,1].
%e A212545 a(11) = 8: [6,1,1,1,1,1], [6,2,1,1,1], [6,2,2,1], [6,3,1,1], [6,4,1], [7,2,1,1], [7,2,2], [8,3].
%p A212545 b:= proc(n, i) option remember;
%p A212545       `if`(n=0 or i=1, 1, b(n, i-1)+`if`(i>n, 0, b(n-i, i)))
%p A212545     end:
%p A212545 a:= n-> add(b(n-2*m-5, min(n-2*m-5, m+5)), m=1..(n-5)/2):
%p A212545 seq(a(n), n=5..60);
%t A212545 b[n_, i_] := b[n, i] = If[n == 0 || i == 1, 1, b[n, i - 1] + If[i > n, 0, b[n - i, i]]];
%t A212545 a[n_] := Sum[b[n - 2 m - 5, Min[n - 2 m - 5, m + 5]], {m, 1, (n - 5)/2}];
%t A212545 a /@ Range[5, 60] (* _Jean-François Alcover_, Dec 07 2020, after _Alois P. Heinz_ *)
%Y A212545 Column k=5 of A212551.
%K A212545 nonn
%O A212545 5,5
%A A212545 _Alois P. Heinz_, May 20 2012