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A300301
Number of ways to choose a partition, with odd parts, of each part of a partition of n into odd parts.
18
1, 1, 1, 3, 3, 6, 10, 15, 21, 37, 56, 80, 127, 183, 280, 428, 616, 893, 1367, 1944, 2846, 4223, 6049, 8691, 12670, 18128, 25921, 37529, 53338, 75738, 108561, 153460, 216762, 308829, 433893, 612006, 864990, 1211097, 1697020, 2386016, 3331037, 4648229, 6503314
OFFSET
0,4
LINKS
FORMULA
O.g.f.: Product_{n odd} 1/(1 - A000009(n)x^n).
EXAMPLE
The a(6) = 10 twice-partitions using odd partitions: (5)(1), (3)(3), (113)(1), (3)(111), (111)(3), (3)(1)(1)(1), (11111)(1), (111)(111), (111)(1)(1)(1), (1)(1)(1)(1)(1)(1).
MAPLE
with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1, add(add(
`if`(d::odd, d, 0), d=divisors(j))*b(n-j), j=1..n)/n)
end:
g:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
g(n, i-2)+`if`(i>n, 0, b(i)*g(n-i, i)))
end:
a:= n-> g(n, n-1+irem(n, 2)):
seq(a(n), n=0..50); # Alois P. Heinz, Mar 05 2018
MATHEMATICA
nn=50;
ser=Product[1/(1-PartitionsQ[n]x^n), {n, 1, nn, 2}];
Table[SeriesCoefficient[ser, {x, 0, n}], {n, 0, nn}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 02 2018
STATUS
approved