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A238864
Number of partitions of n where the difference between consecutive parts is at most 4.
10
1, 1, 2, 3, 5, 7, 11, 14, 20, 26, 36, 46, 63, 79, 104, 131, 169, 210, 269, 332, 418, 515, 640, 782, 967, 1173, 1435, 1736, 2108, 2534, 3062, 3663, 4398, 5243, 6259, 7429, 8834, 10441, 12356, 14559, 17159, 20144, 23661, 27686, 32403, 37807, 44102, 51306, 59680, 69235, 80297, 92924, 107482, 124070, 143157, 164862, 189763, 218057
OFFSET
0,3
COMMENTS
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).
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)
FORMULA
G.f.: 1 + sum(k>=1, q^k/(1-q^k) * prod(i=1..k-1, (1-q^(5*i))/(1-q^i) ) ).
a(n) = Sum_{k=0..4} A238353(n,k). - Alois P. Heinz, Mar 09 2014
a(n) ~ exp(Pi*sqrt(8*n/15)) / (3^(1/4) * 10^(3/4) * n^(3/4)). - Vaclav Kotesovec, Jan 26 2022
MAPLE
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(b(n-i*j, i-1), j=0..min(4, n/i))))
end:
g:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(b(n-i*j, i-1), j=1..n/i)))
end:
a:= n-> add(g(n, k), k=0..n):
seq(a(n), n=0..60); # Alois P. Heinz, Mar 09 2014
MATHEMATICA
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 *)
PROG
(PARI)
N=66; q = 'q + O('q^N);
Vec( 1 + sum(k=1, N, q^k/(1-q^k) * prod(i=1, k-1, (1-q^(5*i))/(1-q^i) ) ) )
CROSSREFS
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).
Sequence in context: A008629 A347572 A363068 * A070289 A035961 A051056
KEYWORD
nonn
AUTHOR
Joerg Arndt, Mar 08 2014
STATUS
approved