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A218510 Number of partitions of n in which any two parts differ by at most 8.

%I A218510 #13 May 20 2018 11:34:52
%S A218510 1,1,2,3,5,7,11,15,22,30,42,55,75,96,127,161,208,260,330,407,509,621,
%T A218510 765,925,1127,1350,1627,1934,2310,2725,3227,3782,4446,5178,6044,7000,
%U A218510 8122,9355,10791,12370,14195,16196,18494,21012,23887,27029,30596,34492,38894
%N A218510 Number of partitions of n in which any two parts differ by at most 8.
%H A218510 Alois P. Heinz, <a href="/A218510/b218510.txt">Table of n, a(n) for n = 0..1000</a>
%F A218510 G.f.: 1 + Sum_{j>0} x^j / Product_{i=0..8} (1-x^(i+j)).
%p A218510 b:= proc(n, i, k) option remember; `if`(n<0 or k<0, 0,
%p A218510       `if`(n=0, 1, `if`(i<1, 0, b(n, i-1, k-1) +b(n-i, i, k))))
%p A218510     end:
%p A218510 a:= n-> `if`(n=0, 1, 0) +add(b(n-i, i, 8), i=1..n):
%p A218510 seq(a(n), n=0..80);
%t A218510 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 A218510 a[n_] := If[n == 0, 1, 0] + Sum[b[n - i, i, 8], {i, 1, n}];
%t A218510 Table[a[n], {n, 0, 80}] (* _Jean-François Alcover_, May 20 2018, after _Alois P. Heinz_ *)
%Y A218510 Column k=8 of A194621.
%K A218510 nonn
%O A218510 0,3
%A A218510 _Alois P. Heinz_, Oct 31 2012