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

0,11

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).

Links

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

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

Example

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

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 *)

Prog

(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

Crossrefs

Column k=1 of A238451.

Cf. A067659, A067661.

Sequence in context: A286745 A286316 A027198 * A237757 A027197 A363221

Adjacent sequences: A238212 A238213 A238214 * A238216 A238217 A238218

Keyword

nonn

Author

Mircea Merca, Feb 20 2014

Extensions

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

Status

approved