A238132 Number of parts in all partitions of n into even number of distinct parts.

0, 0, 0, 2, 2, 4, 4, 6, 6, 8, 12, 14, 18, 24, 32, 38, 50, 60, 76, 90, 110, 134, 162, 190, 228, 270, 322, 380, 446, 524, 616, 720, 838, 980, 1134, 1314, 1526, 1760, 2026, 2336, 2676, 3072, 3518, 4020, 4586, 5232, 5948, 6760, 7676, 8698, 9846, 11142, 12578

Offset

0,4

Links

Alois P. Heinz, Table of n, a(n) for n = 0..5000

Mircea Merca, Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer, Journal of Number Theory, Volume 160, March 2016, Pages 60-75, function s_e(n).

Eric Weisstein's World of Mathematics, q-Polygamma Function, q-Pochhammer Symbol.

Formula

a(n)=(1/2)*A015723(n)-(1/2)*sum{k=0..A235963(n)-1, (-1)^A110654(k)*A000005(n-A001318(k))}=A015723(n)-A238131(n).

G.f.: (1/2)*prod(k>=1, 1+x^k ) * sum(k>=1, x^k/(1+x^k) ) - (1/2)*prod(k>=1, 1-x^k) * sum(k>=1, x^k/(1-x^k) ).

G.f.: -(2 * (x; x)_inf * (log(1-x) + psi_x(1)) + (-1; x)_inf * (log(1-x) + psi_x(1-log(-1)/log(x))))/(4*log(x)), where psi_q(z) is the q-digamma function, (a; q)_inf is the q-Pochhammer symbol, log(-1) = i*Pi. - Vladimir Reshetnikov, Nov 21 2016

a(n) ~ 3^(1/4) * log(2) * exp(Pi*sqrt(n/3)) / (4*Pi*n^(1/4)). - Vaclav Kotesovec, May 27 2018

Example

a(8)=6 because the partitions of 8 into even number of distinct parts are: 7+1, 6+2 and 5+3.

Maple

b:= proc(n, i) option remember; `if`(i*(i+1)/2<n, 0,
     `if`(n=0, [1, 0$3], b(n, i-1)+`if`(i>n, 0, (p->
      [p[2], p[1], p[4]+p[2], p[3]+p[1]])(b(n-i, i-1)))))
    end:
a:= n-> b(n$2)[3]:
seq(a(n), n=0..60);  # Alois P. Heinz, Dec 27 2015

Mathematica

max = 50; s = (1/2)*Product[1+x^k, {k, 1, max}]*Sum[x^k/(1+x^k), {k, 1, max}] - (1/2)*Product[1-x^k, {k, 1, max}]*Sum[x^k/(1-x^k), {k, 1, max}] + O[x]^(max+1); CoefficientList[s, x] (* Jean-François Alcover, Dec 27 2015 *)

Crossrefs

Cf. A000005, A001318, A015723, A110654, A235963.

Sequence in context: A285999 A005186 A259881 * A278296 A332305 A340282

Adjacent sequences: A238129 A238130 A238131 * A238133 A238134 A238135

Keyword

nonn

Author

Mircea Merca, Feb 18 2014

Status

approved