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A212546 Number of partitions of n containing at least one part m-6 if m is the largest part.

%I A212546 #11 Dec 07 2020 09:36:44
%S A212546 0,0,1,1,3,4,8,11,19,26,40,54,79,104,146,190,257,330,436,552,715,896,
%T A212546 1140,1415,1777,2184,2711,3308,4063,4922,5995,7214,8720,10435,12525,
%U A212546 14910,17793,21076,25016,29507,34850,40941,48148,56351,66007,76995,89855,104484
%N A212546 Number of partitions of n containing at least one part m-6 if m is the largest part.
%H A212546 Alois P. Heinz, <a href="/A212546/b212546.txt">Table of n, a(n) for n = 6..1000</a>
%F A212546 G.f.: Sum_{i>0} x^(2*i+6) / Product_{j=1..6+i} (1-x^j).
%e A212546 a(8) = 1: [7,1].
%e A212546 a(9) = 1: [7,1,1].
%e A212546 a(10) = 3: [7,1,1,1], [7,2,1], [8,2].
%e A212546 a(11) = 4: [7,1,1,1,1], [7,2,1,1], [7,3,1], [8,2,1].
%e A212546 a(12) = 8: [7,1,1,1,1,1], [7,2,1,1,1], [7,2,2,1], [7,3,1,1], [7,4,1], [8,2,1,1], [8,2,2], [9,3].
%p A212546 b:= proc(n, i) option remember;
%p A212546       `if`(n=0 or i=1, 1, b(n, i-1)+`if`(i>n, 0, b(n-i, i)))
%p A212546     end:
%p A212546 a:= n-> add(b(n-2*m-6, min(n-2*m-6, m+6)), m=1..(n-6)/2):
%p A212546 seq(a(n), n=6..60);
%t A212546 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 A212546 a[n_] := Sum[b[n - 2 m - 6, Min[n - 2 m - 6, m + 6]], {m, 1, (n - 6)/2}];
%t A212546 a /@ Range[6, 60] (* _Jean-François Alcover_, Dec 07 2020, after _Alois P. Heinz_ *)
%Y A212546 Column k=6 of A212551.
%K A212546 nonn
%O A212546 6,5
%A A212546 _Alois P. Heinz_, May 20 2012