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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

Seiichi Manyama, Table of n, a(n) for n = 0..10000

B. Kim, E. Kim, and J. Lovejoy, Parity bias in partitions, European J. Combin., 89 (2020), 103159, 19 pp.

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):

seq(a(n), n=0..60);  # Alois P. Heinz, Jan 13 2021

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

Cf. A006950, A143184, A340622, A340623.

Sequence in context: A194747 A065423 A239242 * A008733 A244515 A154280

Adjacent sequences:  A340618 A340619 A340620 * A340622 A340623 A340624

KEYWORD

nonn

AUTHOR

Jeremy Lovejoy, Jan 13 2021

STATUS

approved

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Last modified July 29 08:13 EDT 2021. Contains 346340 sequences. (Running on oeis4.)