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Number of partitions of n where the difference between consecutive parts is at most 4.
10

%I #19 Jan 26 2022 06:41:26

%S 1,1,2,3,5,7,11,14,20,26,36,46,63,79,104,131,169,210,269,332,418,515,

%T 640,782,967,1173,1435,1736,2108,2534,3062,3663,4398,5243,6259,7429,

%U 8834,10441,12356,14559,17159,20144,23661,27686,32403,37807,44102,51306,59680,69235,80297,92924,107482,124070,143157,164862,189763,218057

%N Number of partitions of n where the difference between consecutive parts is at most 4.

%C Also the number of partitions of n such that all parts, with the possible exception of the largest are repeated at most four times (by taking conjugates).

%H Vaclav Kotesovec, <a href="/A238864/b238864.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^(5*i))/(1-q^i) ) ).

%F a(n) = Sum_{k=0..4} A238353(n,k). - _Alois P. Heinz_, Mar 09 2014

%F a(n) ~ exp(Pi*sqrt(8*n/15)) / (3^(1/4) * 10^(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(4, 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[4, 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)

%o 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^(5*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), this sequence, A238865 (d=5), 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