cp's OEIS Frontend

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

%I A238133 #50 Feb 16 2025 08:33:21
%S A238133 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,
%T A238133 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,
%U A238133 -1,0,-6,2,2,0,4,3,4,0,2,4,3,0,5,8,2,0,1,-1,1,-3
%N A238133 Difference between A238131(n) and A238132(n).
%C A238133 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.
%C A238133 The convolution of A000005 and A010815.
%H A238133 Alois P. Heinz, <a href="/A238133/b238133.txt">Table of n, a(n) for n = 0..10000</a>
%H A238133 Mircea Merca, <a href="http://dx.doi.org/10.1016/j.jnt.2014.10.009">A new look on the generating function for the number of divisors</a>, Journal of Number Theory, Volume 149, April 2015, Pages 57-69.
%H A238133 Mircea Merca, <a href="http://dx.doi.org/10.1016/j.jnt.2015.08.014">Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer</a>, Journal of Number Theory, Volume 160, March 2016, Pages 60-75, difference s_o(n)-s_e(n).
%H A238133 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/q-PolygammaFunction.html">q-Polygamma Function</a>, <a href="https://mathworld.wolfram.com/q-PochhammerSymbol.html">q-Pochhammer Symbol</a>.
%F A238133 a(n) = Sum_{k=0..A235963(n)-1} (-1)^A110654(k) * A000005(n-A001318(k)).
%F A238133 G.f.: Product_{k>=1} (1-x^k) * Sum_{k>=1} x^k/(1-x^k).
%F A238133 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).
%p A238133 A238133 := proc(n)
%p A238133     add( numtheory[tau](k)*A010815(n-k),k=0..n) ;
%p A238133 end proc: # _R. J. Mathar_, Jun 18 2016
%p A238133 # second Maple program:
%p A238133 b:= proc(n, i) option remember; `if`(i*(i+1)/2<n, 0,
%p A238133      `if`(n=0, [1, 0$3], b(n, i-1)+`if`(i>n, 0, (p->
%p A238133       [p[2], p[1], p[4]+p[2], p[3]+p[1]])(b(n-i, i-1)))))
%p A238133     end:
%p A238133 a:= n-> (p-> p[4]-p[3])(b(n$2)):
%p A238133 seq(a(n), n=0..100);  # _Alois P. Heinz_, Jun 18 2016
%t A238133 Table[SeriesCoefficient[QPochhammer[x] (Log[1 - x] + QPolyGamma[1, x])/Log[x], {x, 0, n}], {n, 0, 80}] (* _Vladimir Reshetnikov_, Nov 20 2016 *)
%Y A238133 Cf. A000005, A001318, A010815, A110654, A235963, A238131, A238132.
%K A238133 sign,look
%O A238133 0,6
%A A238133 _Mircea Merca_, Feb 18 2014