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A212543 Number of partitions of n containing at least one part m-3 if m is the largest part.

%I A212543 #15 Dec 07 2020 09:42:09
%S A212543 0,0,1,1,3,4,8,10,17,22,33,42,60,75,103,128,169,209,271,331,421,513,
%T A212543 642,777,963,1158,1421,1703,2070,2471,2985,3546,4257,5043,6019,7105,
%U A212543 8443,9933,11752,13790,16247,19012,22326,26052,30492,35500,41420,48108,55980
%N A212543 Number of partitions of n containing at least one part m-3 if m is the largest part.
%H A212543 Alois P. Heinz, <a href="/A212543/b212543.txt">Table of n, a(n) for n = 3..1000</a>
%F A212543 G.f.: Sum_{i>0} x^(2*i+3) / Product_{j=1..3+i} (1-x^j).
%e A212543 a(5) = 1: [4,1].
%e A212543 a(6) = 1: [4,1,1].
%e A212543 a(7) = 3: [4,1,1,1], [4,2,1], [5,2].
%e A212543 a(8) = 4: [4,1,1,1,1], [4,2,1,1], [4,3,1], [5,2,1].
%e A212543 a(9) = 8: [4,1,1,1,1,1], [4,2,1,1,1], [4,2,2,1], [4,3,1,1], [4,4,1], [5,2,1,1], [5,2,2], [6,3].
%p A212543 b:= proc(n, i) option remember;
%p A212543       `if`(n=0 or i=1, 1, b(n, i-1)+`if`(i>n, 0, b(n-i, i)))
%p A212543     end:
%p A212543 a:= n-> add(b(n-2*m-3, min(n-2*m-3, m+3)), m=1..(n-3)/2):
%p A212543 seq(a(n), n=3..60);
%t A212543 Table[Count[IntegerPartitions[n],_?(MemberQ[#,#[[1]]-3]&)],{n,3,60}] (* _Harvey P. Dale_, Mar 01 2015 *)
%t A212543 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 A212543 a[n_] := Sum[b[n - 2 m - 3, Min[n - 2 m - 3, m + 3]], {m, 1, (n - 3)/2}];
%t A212543 a /@ Range[3, 60] (* _Jean-François Alcover_, Dec 07 2020, after _Alois P. Heinz_ *)
%Y A212543 Column k=3 of A212551.
%K A212543 nonn
%O A212543 3,5
%A A212543 _Alois P. Heinz_, May 20 2012