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).

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

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