login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A238132 Number of parts in all partitions of n into even number of distinct parts. 6
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 16 15:15 EDT 2022. Contains 353706 sequences. (Running on oeis4.)