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%I #17 Jan 26 2022 08:35:12
%S 1,1,2,3,5,7,11,15,22,29,40,52,71,91,121,155,202,255,328,410,520,647,
%T 810,1000,1244,1525,1879,2293,2804,3401,4135,4988,6028,7241,8701,
%U 10404,12447,14818,17645,20931,24822,29334,34658,40817,48052,56416,66190,77471,90621,105756,123338,143555,166956,193815,224828,260352
%N Number of partitions of n where the difference between consecutive parts is at most 6.
%C Also the number of partitions of n such that all parts, with the possible exception of the largest are repeated at most 6 times (by taking conjugates).
%H Vaclav Kotesovec, <a href="/A238866/b238866.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^(7*i))/(1-q^i) ) ).
%F a(n) = Sum_{k=0..6} A238353(n,k). - _Alois P. Heinz_, Mar 09 2014
%F a(n) ~ exp(Pi*sqrt(4*n/7)) / (2 * 7^(3/4) * 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(6, 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[6, 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 (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^(7*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), A238865 (d=5), this sequence, A238867 (d=7), A238868 (d=8), A238869 (d=9), A000041 (d --> infinity).
%K nonn
%O 0,3
%A _Joerg Arndt_, Mar 08 2014
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