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A218508 Number of partitions of n in which any two parts differ by at most 6.

%I A218508 #13 May 20 2018 11:34:31
%S A218508 1,1,2,3,5,7,11,15,22,29,40,51,69,86,112,139,176,214,268,321,394,470,
%T A218508 567,668,800,933,1103,1281,1498,1725,2006,2293,2643,3010,3443,3897,
%U A218508 4438,4995,5652,6341,7135,7967,8932,9930,11079,12283,13645,15071,16692,18372
%N A218508 Number of partitions of n in which any two parts differ by at most 6.
%H A218508 Alois P. Heinz, <a href="/A218508/b218508.txt">Table of n, a(n) for n = 0..1000</a>
%F A218508 G.f.: 1 + Sum_{j>0} x^j / Product_{i=0..6} (1-x^(i+j)).
%p A218508 b:= proc(n, i, k) option remember; `if`(n<0 or k<0, 0,
%p A218508       `if`(n=0, 1, `if`(i<1, 0, b(n, i-1, k-1) +b(n-i, i, k))))
%p A218508     end:
%p A218508 a:= n-> `if`(n=0, 1, 0) +add(b(n-i, i, 6), i=1..n):
%p A218508 seq(a(n), n=0..80);
%t A218508 b[n_, i_, k_] := b[n, i, k] = If[n < 0 || k < 0, 0, If[n == 0, 1, If[i < 1, 0, b[n, i - 1, k - 1] + b[n - i, i, k]]]];
%t A218508 a[n_] := If[n == 0, 1, 0] + Sum[b[n - i, i, 6], {i, 1, n}];
%t A218508 Table[a[n], {n, 0, 80}] (* _Jean-François Alcover_, May 20 2018, after _Alois P. Heinz_ *)
%Y A218508 Column k=6 of A194621.
%K A218508 nonn
%O A218508 0,3
%A A218508 _Alois P. Heinz_, Oct 31 2012