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%I #13 Dec 07 2020 09:06:28
%S 0,0,1,1,3,4,8,11,19,26,41,56,82,110,156,205,281,366,488,627,821,1041,
%T 1340,1684,2135,2657,3331,4108,5095,6238,7663,9315,11354,13709,16588,
%U 19915,23936,28580,34154,40573,48225,57031,67452,79428,93530,109695,128639
%N Number of partitions of n containing at least one part m-8 if m is the largest part.
%H Alois P. Heinz, <a href="/A212548/b212548.txt">Table of n, a(n) for n = 8..1000</a>
%F G.f.: Sum_{i>0} x^(2*i+8) / Product_{j=1..8+i} (1-x^j).
%e a(10) = 1: [9,1].
%e a(11) = 1: [9,1,1].
%e a(12) = 3: [9,1,1,1], [9,2,1], [10,2].
%e a(13) = 4: [9,1,1,1,1], [9,2,1,1], [9,3,1], [10,2,1].
%e a(14) = 8: [9,1,1,1,1,1], [9,2,1,1,1], [9,2,2,1], [9,3,1,1], [9,4,1], [10,2,1,1], [10,2,2], [11,3].
%p b:= proc(n, i) option remember;
%p `if`(n=0 or i=1, 1, b(n, i-1)+`if`(i>n, 0, b(n-i, i)))
%p end:
%p a:= n-> add(b(n-2*m-8, min(n-2*m-8, m+8)), m=1..(n-8)/2):
%p seq(a(n), n=8..60);
%t Table[Count[IntegerPartitions[n],_?(MemberQ[#,Max[#]-8]&)],{n,8,55}] (* _Harvey P. Dale_, May 05 2016 *)
%t 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 a[n_] := Sum[b[n - 2 m - 8, Min[n - 2 m - 8, m + 8]], {m, 1, (n - 8)/2}];
%t a /@ Range[8, 60] (* _Jean-François Alcover_, Dec 07 2020, after _Alois P. Heinz_ *)
%Y Column k=8 of A212551.
%K nonn
%O 8,5
%A _Alois P. Heinz_, May 20 2012
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