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A282893 The difference between the number of partitions of 2n into odd parts (A000009) and the number of partitions of 2n into even parts (A035363). 3
0, 0, 0, 1, 1, 3, 4, 7, 10, 16, 22, 33, 45, 64, 87, 120, 159, 215, 283, 374, 486, 634, 814, 1049, 1335, 1700, 2146, 2708, 3390, 4243, 5276, 6552, 8095, 9989, 12266, 15044, 18375, 22409, 27235, 33049, 39974, 48281, 58148, 69923, 83871, 100452, 120027, 143214, 170515, 202731, 240567, 285073, 337195 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

The even bisection of A282892. The other bisection is A078408.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

a(n) = A282892(2n).

Expansion of (f(x^3, x^5) - 1) / f(-x, -x^2) in powers of x where f(, ) is Ramanujan's general theta function. - Michael Somos, Feb 24 2017

a(n) = A035294(n) - A000041(n). - Michael Somos, Feb 24 2017

EXAMPLE

G.f. = x^3 + x^4 + 3*x^5 + 4*x^6 + 7*x^7 + 10*x^8 + 16*x^9 + 22*x^10 + 33*x^11 + ...

MAPLE

with(numtheory):

b:= proc(n, t) option remember; `if`(n=0, 1, add(add(`if`(

      (d+t)::odd, d, 0), d=divisors(j))*b(n-j, t), j=1..n)/n)

    end:

a:= n-> b(2*n, 0) -b(2*n, 1):

seq(a(n), n=0..80);  # Alois P. Heinz, Feb 24 2017

MATHEMATICA

f[n_] := Length[ IntegerPartitions[n, All, 2Range[n] -1]] - Length[ IntegerPartitions[n, All, 2 Range[n]]]; Array[ f[2#] &, 52]

a[ n_] := SeriesCoefficient[ Sum[ Sign @ SquaresR[1, 16 k + 1] x^k, {k, n}] / QPochhammer[x], {x, 0, n}]; (* Michael Somos, Feb 24 2017 *)

PROG

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( sum(k=1, n, issquare(16*k + 1)*x^k, A) / eta(x + A), n))}; /* Michael Somos, Feb 24 2017 */

CROSSREFS

Cf. A000009, A000041, A035294, A035363, A078408, A214264, A282892.

Sequence in context: A324368 A241654 A047967 * A256912 A134591 A058611

Adjacent sequences:  A282890 A282891 A282892 * A282894 A282895 A282896

KEYWORD

nonn

AUTHOR

Robert G. Wilson v, Feb 24 2017

STATUS

approved

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Last modified June 19 14:08 EDT 2021. Contains 345140 sequences. (Running on oeis4.)