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A218507 Number of partitions of n in which any two parts differ by at most 5.

%I A218507 #13 Jan 18 2016 15:37:09
%S A218507 1,1,2,3,5,7,11,15,21,28,37,48,62,78,98,121,149,181,219,262,313,370,
%T A218507 436,510,595,690,797,916,1050,1198,1364,1545,1747,1968,2212,2479,2771,
%U A218507 3089,3437,3814,4226,4669,5151,5670,6232,6837,7487,8185,8936,9739,10602
%N A218507 Number of partitions of n in which any two parts differ by at most 5.
%H A218507 Alois P. Heinz, <a href="/A218507/b218507.txt">Table of n, a(n) for n = 0..1000</a>
%H A218507 <a href="/index/Rec#order_20">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,0,0,-2,-2,1,1,2,1,1,-2,-2,0,0,0,1,1,-1).
%F A218507 G.f.: 1 + Sum_{j>0} x^j / Product_{i=0..5} (1-x^(i+j)).
%F A218507 G.f.: (x^20 -x^19 -x^18 +x^15 +x^14 +x^13 -x^12 -x^11 -x^10 +x^7 +x^6 -x^5 +1) / ((x -1)^6*(x +1)^2*(x^2 +1)*(x^2 +x +1)*(x^4 +x^3 +x^2 +x +1)^2). - _Colin Barker_, Mar 05 2015
%p A218507 b:= proc(n, i, k) option remember; `if`(n<0 or k<0, 0,
%p A218507       `if`(n=0, 1, `if`(i<1, 0, b(n, i-1, k-1) +b(n-i, i, k))))
%p A218507     end:
%p A218507 a:= n-> `if`(n=0, 1, 0) +add(b(n-i, i, 5), i=1..n):
%p A218507 seq(a(n), n=0..80);
%t A218507 LinearRecurrence[{1,1,0,0,0,-2,-2,1,1,2,1,1,-2,-2,0,0,0,1,1,-1},{1,1,2,3,5,7,11,15,21,28,37,48,62,78,98,121,149,181,219,262,313},60] (* _Harvey P. Dale_, Jan 18 2016 *)
%o A218507 (PARI) Vec((x^20-x^19-x^18+x^15+x^14+x^13-x^12-x^11-x^10+x^7+x^6-x^5+1)/((x-1)^6*(x+1)^2*(x^2+1)*(x^2+x+1)*(x^4+x^3+x^2+x+1)^2) + O(x^100)) \\ _Colin Barker_, Mar 05 2015
%Y A218507 Column k=5 of A194621.
%K A218507 nonn,easy
%O A218507 0,3
%A A218507 _Alois P. Heinz_, Oct 31 2012