A238133 Difference between A238131(n) and A238132(n).

0, 1, 1, -1, -1, -3, 0, -2, 1, 2, 1, 2, 4, 1, -1, 4, -2, -1, -3, -1, -2, -2, -6, 0, -1, 1, -4, 0, 3, 2, 2, 2, 3, 0, 4, 7, 0, 0, 2, -3, 7, -2, -1, -3, -2, -4, 0, -3, -3, -2, -1, -10, -1, 0, 1, -1, 0, -6, 2, 2, 0, 4, 3, 4, 0, 2, 4, 3, 0, 5, 8, 2, 0, 1, -1, 1, -3

Offset: 0

Author

Mircea Merca, Feb 18 2014

Keywords

• sign
• look

Comments

Difference between the number of parts in all partitions of n into odd number of distinct parts and the number of parts in all partitions of n into even number of distinct parts.

The convolution of A000005 and A010815.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000
• Mircea Merca, A new look on the generating function for the number of divisors, Journal of Number Theory, Volume 149, April 2015, Pages 57-69.
• 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, difference s_o(n)-s_e(n).
• Eric Weisstein's World of Mathematics, q-Polygamma Function, q-Pochhammer Symbol.

Crossrefs

Cf. A000005, A001318, A010815, A110654, A235963, A238131, A238132.

Programs

Maple

A238133 := proc(n)
    add( numtheory[tau](k)*A010815(n-k),k=0..n) ;
end proc: # R. J. Mathar, Jun 18 2016
# second Maple program:
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-> (p-> p[4]-p[3])(b(n$2)):
seq(a(n), n=0..100);  # Alois P. Heinz, Jun 18 2016

Mathematica

Table[SeriesCoefficient[QPochhammer[x] (Log[1 - x] + QPolyGamma[1, x])/Log[x], {x, 0, n}], {n, 0, 80}] (* Vladimir Reshetnikov, Nov 20 2016 *)

Formula

a(n) = Sum_{k=0..A235963(n)-1} (-1)^A110654(k) * A000005(n-A001318(k)).

G.f.: Product_{k>=1} (1-x^k) * Sum_{k>=1} x^k/(1-x^k).

G.f.: (x)_inf * (log(1-x) + psi_x(1))/log(x), where psi_q(z) is the q-digamma function, (q)_inf is the q-Pochhammer symbol (the Euler function).