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A218509 Number of partitions of n in which any two parts differ by at most 7.

%I A218509 #13 May 20 2018 11:34:42
%S A218509 1,1,2,3,5,7,11,15,22,30,41,54,72,93,120,153,194,242,302,372,457,556,
%T A218509 675,812,975,1162,1381,1632,1923,2254,2636,3068,3562,4119,4752,5462,
%U A218509 6265,7162,8170,9293,10549,11942,13495,15211,17115,19214,21534,24083,26892
%N A218509 Number of partitions of n in which any two parts differ by at most 7.
%H A218509 Alois P. Heinz, <a href="/A218509/b218509.txt">Table of n, a(n) for n = 0..1000</a>
%F A218509 G.f.: 1 + Sum_{j>0} x^j / Product_{i=0..7} (1-x^(i+j)).
%p A218509 b:= proc(n, i, k) option remember; `if`(n<0 or k<0, 0,
%p A218509       `if`(n=0, 1, `if`(i<1, 0, b(n, i-1, k-1) +b(n-i, i, k))))
%p A218509     end:
%p A218509 a:= n-> `if`(n=0, 1, 0) +add(b(n-i, i, 7), i=1..n):
%p A218509 seq(a(n), n=0..80);
%t A218509 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 A218509 a[n_] := If[n == 0, 1, 0] + Sum[b[n - i, i, 7], {i, 1, n}];
%t A218509 Table[a[n], {n, 0, 80}] (* _Jean-François Alcover_, May 20 2018, after _Alois P. Heinz_ *)
%Y A218509 Column k=7 of A194621.
%K A218509 nonn
%O A218509 0,3
%A A218509 _Alois P. Heinz_, Oct 31 2012