|
|
A238863
|
|
Number of partitions of n where the difference between consecutive parts is at most 3.
|
|
10
|
|
|
1, 1, 2, 3, 5, 7, 10, 13, 18, 24, 32, 41, 54, 68, 87, 111, 139, 174, 218, 269, 333, 410, 501, 611, 745, 902, 1090, 1315, 1578, 1891, 2263, 2695, 3205, 3805, 4503, 5322, 6277, 7384, 8673, 10172, 11904, 13908, 16227, 18894, 21971, 25516, 29578, 34245, 39597, 45717, 52720, 60721, 69842, 80243, 92091, 105559, 120865, 138248
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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 three times (by taking conjugates).
The g.f. for "max difference d" is 1 + Sum_{k>=1} q^k/(1 - q^k) * Product_{i=1..k-1} (1 - q^((d+1)*i))/(1 - q^i), see cross references.
|
|
LINKS
|
|
|
FORMULA
|
G.f.: 1 + Sum_{k>=1} (q^k/(1 - q^k)) * Product_{i=1..k-1} (1 - q^(4*i))/(1 - q^i).
a(n) ~ exp(Pi*sqrt(n/2)) / (2^(11/4) * 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(3, 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):
|
|
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[3, 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^(4*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), this sequence, A238864 (d=4), A238865 (d=5), A238866 (d=6), A238867 (d=7), A238868 (d=8), A238869 (d=9), A000041 (d --> infinity).
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|