A317910 Expansion of -1/(1 - x)^2 + (1/(1 - x))*Product_{k>=1} (1 + x^k).

0, 0, 0, 1, 2, 4, 7, 11, 16, 23, 32, 43, 57, 74, 95, 121, 152, 189, 234, 287, 350, 425, 513, 616, 737, 878, 1042, 1233, 1454, 1709, 2004, 2343, 2732, 3179, 3690, 4274, 4941, 5700, 6563, 7544, 8656, 9915, 11340, 12949, 14764, 16811, 19114, 21703, 24612, 27875, 31532, 35628, 40209

Offset: 0

Author

Ilya Gutkovskiy, Aug 10 2018

Keywords

• nonn

Comments

Partial sums of A111133.

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..10000
• Riccardo Aragona, Roberto Civino, and Norberto Gavioli, A modular idealizer chain and unrefinability of partitions with repeated parts, arXiv:2301.06347 [math.RA], 2023.
• Riccardo Aragona, Roberto Civino, Norberto Gavioli, Carlo Maria Scoppola, A Chain of Normalizers in the Sylow _2-subgroups of the symmetric group on 2^n letters, arXiv:2008.13423 [math.GR], 2020.
• Riccardo Aragona, Roberto Civino, Norberto Gavioli, and Carlo Maria Scoppola, Rigid commutators and a normalizer chain, arXiv:2009.11149 [math.GR], 2020.
• Index entries for sequences related to partitions

Crossrefs

Cf. A026906, A036469, A058682, A111133.

Programs

Maple

a:=series(-1/(1-x)^2+(1/(1-x))*mul((1 + x^k),k=1..100),x=0,53): seq(coeff(a,x,n),n=0..52); # Paolo P. Lava, Apr 02 2019

Mathematica

nmax = 52; CoefficientList[Series[-1/(1 - x)^2 + 1/(1 - x) Product[1 + x^k, {k, 1, nmax}], {x, 0, nmax}], x] (* or *)
nmax = 52; CoefficientList[Series[1/((1 - x) QPochhammer[x, x^2]) - 1/(1 - x)^2, {x, 0, nmax}], x] (* or *)
Table[Sum[PartitionsQ[k] - 1, {k, 0, n}] , {n, 0, 52}]

Formula

G.f.: -1/(1 - x)^2 + (1/(1 - x))*Product_{k>=1} 1/(1 - x^(2*k-1)).

a(n) = A036469(n) - n - 1.

a(n) ~ 3^(1/4) * exp(Pi*sqrt(n/3)) / (2*Pi*n^(1/4)). - Vaclav Kotesovec, Aug 21 2018