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 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)/2n, 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

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