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A238863 Number of partitions of n where the difference between consecutive parts is at most 3.

%I A238863 #22 Feb 24 2022 02:07:16
%S A238863 1,1,2,3,5,7,10,13,18,24,32,41,54,68,87,111,139,174,218,269,333,410,
%T A238863 501,611,745,902,1090,1315,1578,1891,2263,2695,3205,3805,4503,5322,
%U A238863 6277,7384,8673,10172,11904,13908,16227,18894,21971,25516,29578,34245,39597,45717,52720,60721,69842,80243,92091,105559,120865,138248
%N A238863 Number of partitions of n where the difference between consecutive parts is at most 3.
%C A238863 Also the number of partitions of n such that all parts, with the possible exception of the largest are repeated at most three times (by taking conjugates).
%C A238863 The g.f. for "max difference d" is 1 + Sum_{k>=1} q^k/(1 - q^k) * Product_{i=1..k-1} (1 - q^((d+1)*i))/(1 - q^i), see cross references.
%H A238863 Vaclav Kotesovec, <a href="/A238863/b238863.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Alois P. Heinz)
%F A238863 G.f.: 1 + Sum_{k>=1} (q^k/(1 - q^k)) * Product_{i=1..k-1} (1 - q^(4*i))/(1 - q^i).
%F A238863 a(n) = Sum_{k=0..3} A238353(n,k). - _Alois P. Heinz_, Mar 09 2014
%F A238863 a(n) ~ exp(Pi*sqrt(n/2)) / (2^(11/4) * n^(3/4)). - _Vaclav Kotesovec_, Jan 26 2022
%p A238863 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A238863       add(b(n-i*j, i-1), j=0..min(3, n/i))))
%p A238863     end:
%p A238863 g:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A238863       add(b(n-i*j, i-1), j=1..n/i)))
%p A238863     end:
%p A238863 a:= n-> add(g(n, k), k=0..n):
%p A238863 seq(a(n), n=0..60);  # _Alois P. Heinz_, Mar 09 2014
%t A238863 b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[b[n-i*j, i-1], {j, 0, Min[3, 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_ *)
%o A238863 (PARI)
%o A238863 N=66;  q = 'q + O('q^N);
%o A238863 Vec( 1 + sum(k=1, N, q^k/(1-q^k) * prod(i=1,k-1, (1-q^(4*i))/(1-q^i) ) ) )
%Y A238863 Sequences "number of partitions with max diff d": A000005 (d=0, for n>=1), A034296 (d=1), A224956 (d=2), this sequence, A238864 (d=4), A238865 (d=5), A238866 (d=6), A238867 (d=7), A238868 (d=8), A238869 (d=9), A000041 (d --> infinity).
%K A238863 nonn
%O A238863 0,3
%A A238863 _Joerg Arndt_, Mar 08 2014