A238215 The total number of 1's in all partitions of n into an even number of distinct parts.

0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 9, 11, 13, 15, 18, 21, 24, 28, 33, 38, 44, 51, 59, 68, 79, 90, 104, 119, 136, 156, 178, 202, 230, 261, 296, 335, 379, 427, 482, 543, 610, 686, 770, 863, 967, 1082, 1209, 1351, 1508, 1681, 1873, 2085, 2318, 2577

Offset: 0

Author

Mircea Merca, Feb 20 2014

Keywords

• nonn

Comments

The g.f. for "number of k's" is (1/2)*(x^k/(1+x^k))*(Product_{n>=1} 1 + x^n) - (1/2)*(x^k/(1-x^k))*(Product_{n>=1} 1 - x^n).

Examples

			a(12) = 3 because the partitions in question are: 11+1, 6+3+2+1, 5+4+2+1.
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1000

Crossrefs

Column k=1 of A238451.

Cf. A067659, A067661.

Programs

Maple

b:= proc(n, i, t) option remember; `if`(n=0, t,
     `if`(i>n, 0, b(n, i+1, t)+b(n-i, i+1, 1-t)))
    end:
a:= n-> b(n-1, 2, 0):
seq(a(n), n=0..100);  # Alois P. Heinz, May 01 2020

Mathematica

b[n_, i_, t_] := b[n, i, t] = If[n == 0, t, If[i > n, 0, b[n, i + 1, t] + b[n - i, i + 1, 1 - t]]];
a[n_] := b[n - 1, 2, 0];
a /@ Range[0, 100] (* Jean-François Alcover, Nov 01 2020, after Alois P. Heinz *)

PARI

seq(n)={my(A=O(x^n)); Vec(x*(eta(x^2 + A)/(eta(x + A)*(1+x)) - eta(x + A)/(1-x))/2, -(n+1))} \\ Andrew Howroyd, May 01 2020

Formula

a(n) = Sum_{j=1..round(n/2)} A067659(n-(2*j-1)) - Sum_{j=1..floor(n/2)} A067661(n-2*j).

G.f.: (1/2)*(x/(1+x))*(Product_{n>=1} 1 + x^n) - (1/2)*(x/(1-x))*(Product_{n>=1} 1 - x^n).

a(n) ~ exp(Pi*sqrt(n/3)) / (16*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Nov 07 2024

Extensions

Terms a(51) and beyond from Andrew Howroyd, May 01 2020