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A212544 Number of partitions of n containing at least one part m-4 if m is the largest part.

%I A212544 #13 Dec 07 2020 08:10:41
%S A212544 0,0,1,1,3,4,8,11,18,24,37,48,69,89,122,155,207,259,337,419,534,657,
%T A212544 827,1008,1252,1518,1864,2246,2736,3276,3960,4722,5668,6727,8032,9492,
%U A212544 11274,13279,15696,18424,21694,25380,29772,34736,40604,47244,55060,63897
%N A212544 Number of partitions of n containing at least one part m-4 if m is the largest part.
%H A212544 Alois P. Heinz, <a href="/A212544/b212544.txt">Table of n, a(n) for n = 4..1000</a>
%F A212544 G.f.: Sum_{i>0} x^(2*i+4) / Product_{j=1..4+i} (1-x^j).
%e A212544 a(6) = 1: [5,1].
%e A212544 a(7) = 1: [5,1,1].
%e A212544 a(8) = 3: [5,1,1,1], [5,2,1], [6,2].
%e A212544 a(9) = 4: [5,1,1,1,1], [5,2,1,1], [5,3,1], [6,2,1].
%e A212544 a(10) = 8: [5,1,1,1,1,1], [5,2,1,1,1], [5,2,2,1], [5,3,1,1], [5,4,1], [6,2,1,1], [6,2,2], [7,3].
%p A212544 b:= proc(n, i) option remember;
%p A212544       `if`(n=0 or i=1, 1, b(n, i-1)+`if`(i>n, 0, b(n-i, i)))
%p A212544     end:
%p A212544 a:= n-> add(b(n-2*m-4, min(n-2*m-4, m+4)), m=1..(n-4)/2):
%p A212544 seq(a(n), n=4..60);
%t A212544 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 A212544 a[n_] := Sum[b[n - 2 m - 4, Min[n - 2 m - 4, m + 4]], {m, 1, (n - 4)/2}];
%t A212544 a /@ Range[4, 60] (* _Jean-François Alcover_, Dec 07 2020, after _Alois P. Heinz_ *)
%Y A212544 Column k=4 of A212551.
%K A212544 nonn
%O A212544 4,5
%A A212544 _Alois P. Heinz_, May 20 2012