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A218512 Number of partitions of n in which any two parts differ by at most 10.

%I A218512 #16 Oct 29 2024 10:55:38
%S A218512 1,1,2,3,5,7,11,15,22,30,42,56,77,100,133,171,223,282,362,453,572,709,
%T A218512 884,1084,1337,1626,1984,2394,2896,3468,4162,4951,5897,6972,8249,9696,
%U A218512 11402,13330,15586,18131,21090,24417,28264,32580,37541,43097,49449,56544
%N A218512 Number of partitions of n in which any two parts differ by at most 10.
%H A218512 Alois P. Heinz, <a href="/A218512/b218512.txt">Table of n, a(n) for n = 0..2000</a>
%F A218512 G.f.: 1 + Sum_{j>0} x^j / Product_{i=0..10} (1-x^(i+j)).
%p A218512 b:= proc(n, i, k) option remember; `if`(n<0 or k<0, 0,
%p A218512       `if`(n=0, 1, `if`(i<1, 0, b(n, i-1, k-1) +b(n-i, i, k))))
%p A218512     end:
%p A218512 a:= n-> `if`(n=0, 1, 0) +add(b(n-i, i, 10), i=1..n):
%p A218512 seq(a(n), n=0..80);
%t A218512 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 A218512 a[n_] := If[n == 0, 1, 0] + Sum[b[n - i, i, 10], {i, 1, n}];
%t A218512 Table[a[n], {n, 0, 80}] (* _Jean-François Alcover_, May 21 2018, translated from Maple *)
%Y A218512 Column k=10 of A194621.
%K A218512 nonn
%O A218512 0,3
%A A218512 _Alois P. Heinz_, Oct 31 2012