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A238865
Number of partitions of n where the difference between consecutive parts is at most 5.
10
1, 1, 2, 3, 5, 7, 11, 15, 21, 28, 38, 50, 67, 87, 114, 146, 188, 238, 302, 379, 476, 593, 737, 911, 1124, 1379, 1688, 2058, 2504, 3034, 3669, 4422, 5319, 6378, 7634, 9114, 10859, 12908, 15316, 18134, 21434, 25283, 29775, 35001, 41080, 48133, 56312, 65778, 76727, 89366, 103947, 120739, 140065, 162271, 187769, 217006, 250504
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 five 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^(6*i))/(1-q^i) ) ).
a(n) = Sum_{k=0..5} A238353(n,k). - Alois P. Heinz, Mar 09 2014
a(n) ~ 5^(1/4) * exp(Pi*sqrt(5*n/9)) / (12 * 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(5, 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[5, 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 *)
Table[Count[IntegerPartitions[n], _?(Max[Abs[Differences[#]]]<6&)], {n, 0, 60}] (* Harvey P. Dale, Feb 04 2017 *)
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^(6*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), A238864 (d=4), this sequence, A238866 (d=6), A238867 (d=7), A238868 (d=8), A238869 (d=9), A000041 (d --> infinity).
Sequence in context: A377075 A008630 A347573 * A326978 A035969 A355027
KEYWORD
nonn
AUTHOR
Joerg Arndt, Mar 08 2014
STATUS
approved