%I #19 Jan 26 2022 08:33:03
%S 1,1,2,3,5,7,11,15,21,28,38,50,67,87,114,146,188,238,302,379,476,593,
%T 737,911,1124,1379,1688,2058,2504,3034,3669,4422,5319,6378,7634,9114,
%U 10859,12908,15316,18134,21434,25283,29775,35001,41080,48133,56312,65778,76727,89366,103947,120739,140065,162271,187769,217006,250504
%N Number of partitions of n where the difference between consecutive parts is at most 5.
%C Also the number of partitions of n such that all parts, with the possible exception of the largest are repeated at most five times (by taking conjugates).
%H Vaclav Kotesovec, <a href="/A238865/b238865.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Alois P. Heinz)
%F G.f.: 1 + sum(k>=1, q^k/(1-q^k) * prod(i=1..k-1, (1-q^(6*i))/(1-q^i) ) ).
%F a(n) = Sum_{k=0..5} A238353(n,k). - _Alois P. Heinz_, Mar 09 2014
%F a(n) ~ 5^(1/4) * exp(Pi*sqrt(5*n/9)) / (12 * n^(3/4)). - _Vaclav Kotesovec_, Jan 26 2022
%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p add(b(n-i*j, i-1), j=0..min(5, n/i))))
%p end:
%p g:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p add(b(n-i*j, i-1), j=1..n/i)))
%p end:
%p a:= n-> add(g(n, k), k=0..n):
%p seq(a(n), n=0..60); # _Alois P. Heinz_, Mar 09 2014
%t b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[b[n - i*j, i-1], {j, 0, Min[5, n/i]}]]]; g[n_, i_] := g[n, i] = If[n == 0, 1, If[i<1, 0, Sum[b[n - i*j, i-1], {j, 1, n/i}]]]; a[n_] := Sum[g[n, k], {k, 0, n}]; Table[a[n], {n, 0, 60}] (* _Jean-François Alcover_, Feb 18 2015, after _Alois P. Heinz_ *)
%t Table[Count[IntegerPartitions[n],_?(Max[Abs[Differences[#]]]<6&)],{n,0,60}] (* _Harvey P. Dale_, Feb 04 2017 *)
%o (PARI) N=66; q = 'q + O('q^N);
%o Vec( 1 + sum(k=1, N, q^k/(1-q^k) * prod(i=1,k-1, (1-q^(6*i))/(1-q^i) ) ) )
%Y Sequences "number of partitions with max diff d": A000005 (d=0, for n>=1), A034296 (d=1), A224956 (d=2), A238863 (d=3), A238864 (d=4), this sequence, A238866 (d=6), A238867 (d=7), A238868 (d=8), A238869 (d=9), A000041 (d --> infinity).
%K nonn
%O 0,3
%A _Joerg Arndt_, Mar 08 2014