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%I #18 Jan 26 2022 09:29:41
%S 1,1,2,3,5,7,11,15,22,30,42,55,75,97,129,166,217,276,356,449,572,715,
%T 900,1117,1393,1717,2123,2601,3193,3889,4744,5748,6970,8404,10135,
%U 12165,14600,17448,20845,24813,29522,35009,41491,49031,57900,68195,80258,94234,110553,129421,151382,176724,206132,240002,279195,324255
%N Number of partitions of n where the difference between consecutive parts is at most 8.
%C Also the number of partitions of n such that all parts, with the possible exception of the largest are repeated at most eight times (by taking conjugates).
%H Vaclav Kotesovec, <a href="/A238868/b238868.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^(9*i))/(1-q^i) ) ).
%F a(n) = Sum_{k=0..8} A238353(n,k). - _Alois P. Heinz_, Mar 09 2014
%F a(n) ~ exp(4*Pi*sqrt(n/27)) / (sqrt(2) * 3^(7/4) * 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(8, 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[8, 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^(9*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), A238866 (d=6), A238867 (d=7), this sequence, A238869 (d=9), A000041 (d --> infinity).
%K nonn
%O 0,3
%A _Joerg Arndt_, Mar 08 2014
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