cp's OEIS Frontend

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).

%I A282893 #24 Feb 16 2025 08:33:42
%S A282893 0,0,0,1,1,3,4,7,10,16,22,33,45,64,87,120,159,215,283,374,486,634,814,
%T A282893 1049,1335,1700,2146,2708,3390,4243,5276,6552,8095,9989,12266,15044,
%U A282893 18375,22409,27235,33049,39974,48281,58148,69923,83871,100452,120027,143214,170515,202731,240567,285073,337195
%N 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).
%C A282893 The even bisection of A282892. The other bisection is A078408.
%H A282893 Alois P. Heinz, <a href="/A282893/b282893.txt">Table of n, a(n) for n = 0..10000</a>
%H A282893 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A282893 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A282893 a(n) = A282892(2n).
%F A282893 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
%F A282893 a(n) = A035294(n) - A000041(n). - _Michael Somos_, Feb 24 2017
%e A282893 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 + ...
%p A282893 with(numtheory):
%p A282893 b:= proc(n, t) option remember; `if`(n=0, 1, add(add(`if`(
%p A282893       (d+t)::odd, d, 0), d=divisors(j))*b(n-j, t), j=1..n)/n)
%p A282893     end:
%p A282893 a:= n-> b(2*n, 0) -b(2*n, 1):
%p A282893 seq(a(n), n=0..80);  # _Alois P. Heinz_, Feb 24 2017
%t A282893 f[n_] := Length[ IntegerPartitions[n, All, 2Range[n] -1]] - Length[ IntegerPartitions[n, All, 2 Range[n]]]; Array[ f[2#] &, 52]
%t A282893 a[ n_] := SeriesCoefficient[ Sum[ Sign @ SquaresR[1, 16 k + 1] x^k, {k, n}] / QPochhammer[x], {x, 0, n}]; (* _Michael Somos_, Feb 24 2017 *)
%o A282893 (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 */
%Y A282893 Cf. A000009, A000041, A035294, A035363, A078408, A214264, A282892.
%K A282893 nonn
%O A282893 0,6
%A A282893 _Robert G. Wilson v_, Feb 24 2017