|
|
A340621
|
|
The number of partitions of n without repeated odd parts having more odd parts than even parts.
|
|
3
|
|
|
0, 1, 0, 1, 1, 1, 2, 1, 4, 2, 6, 3, 9, 6, 12, 10, 17, 17, 22, 26, 30, 40, 40, 57, 55, 82, 74, 112, 103, 153, 140, 203, 193, 270, 262, 351, 357, 458, 478, 589, 641, 760, 846, 971, 1114, 1244, 1450, 1582, 1880, 2018, 2412, 2558, 3086, 3247, 3914, 4102, 4949
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,7
|
|
LINKS
|
|
|
FORMULA
|
G.f.: Sum_{n>=1} q^(n^2)*(1-q^(2*n))/Product_{k=1..n} (1-q^(2*k))^2.
a(n) ~ exp(Pi*sqrt(2*n/5)) / (2^(3/2) * 5^(3/4) * n). - Vaclav Kotesovec, Jan 14 2021
|
|
EXAMPLE
|
a(8) = 4 counts the partitions [7,1], [5,3], [5,2,1], and [4,3,1].
|
|
MAPLE
|
b:= proc(n, i, c) option remember; `if`(n=0,
`if`(c>0, 1, 0), `if`(i<1, 0, (t-> add(b(n-i*j, i-1, c+j*
`if`(t, 1, -1)), j=0..min(n/i, `if`(t, 1, n))))(i::odd)))
end:
a:= n-> b(n$2, 0):
|
|
MATHEMATICA
|
b[n_, i_, c_] := b[n, i, c] = If[n == 0,
If[c > 0, 1, 0], If[i < 1, 0, Function[t, Sum[b[n - i*j, i - 1, c + j*
If[t, 1, -1]], {j, 0, Min[n/i, If[t, 1, n]]}]][OddQ[i]]]];
a[n_] := b[n, n, 0];
|
|
PROG
|
(PARI) my(N=66, x='x+O('x^N)); concat(0, Vec(sum(k=1, sqrt(N), x^(k^2)*(1-x^(2*k))/prod(j=1, k, (1-x^(2*j))^2)))) \\ Seiichi Manyama, Jan 14 2021
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|