A238131 Number of parts in all partitions of n into odd number of distinct parts.

0, 1, 1, 1, 1, 1, 4, 4, 7, 10, 13, 16, 22, 25, 31, 42, 48, 59, 73, 89, 108, 132, 156, 190, 227, 271, 318, 380, 449, 526, 618, 722, 841, 980, 1138, 1321, 1526, 1760, 2028, 2333, 2683, 3070, 3517, 4017, 4584, 5228, 5948, 6757, 7673, 8696, 9845, 11132, 12577

Offset

0,7

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..5000 from Alois P. Heinz)

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_o(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))}.

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)=7 because the partitions of 8 into odd number of distinct parts are: 8, 5+2+1 and 4+3+1.

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)[4]:
seq(a(n), n=0..50);  # 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, A079499.

Sequence in context: A214990 A185670 A011981 * A212532 A109544 A187893

Adjacent sequences: A238128 A238129 A238130 * A238132 A238133 A238134

Keyword

nonn

Author

Mircea Merca, Feb 18 2014

Status

approved