A320942 Expansion of Sum_{k>=1} (-1 + Product_{j>=1} (1 + x^(k*j))/(1 - x^(k*j))).

2, 6, 10, 20, 26, 54, 66, 120, 164, 262, 346, 572, 730, 1110, 1506, 2182, 2866, 4156, 5402, 7612, 9978, 13638, 17730, 24200, 31092, 41558, 53572, 70692, 90250, 118406, 150146, 194794, 246610, 316678, 398730, 509560, 637594, 808342, 1009186, 1270984, 1578530, 1978758, 2447066

Offset: 1

Author

Ilya Gutkovskiy, Oct 24 2018

Keywords

• nonn

Comments

Inverse Möbius transform of A015128.

Links

• N. J. A. Sloane, Transforms

Crossrefs

Cf. A015128, A047966, A047968, A300274.

Programs

Maple

a:=series(add(-1+mul((1+x^(k*j))/(1-x^(k*j)),j=1..100),k=1..100),x=0,44): seq(coeff(a,x,n),n=1..43); # Paolo P. Lava, Apr 02 2019

Mathematica

nmax = 43; Rest[CoefficientList[Series[Sum[-1 + Product[(1 + x^(k j))/(1 - x^(k j)), {j, 1, nmax}], {k, 1, nmax}], {x, 0, nmax}], x]]
nmax = 43; Rest[CoefficientList[Series[Sum[1/EllipticTheta[4, 0, x^k] - 1, {k, 1, nmax}], {x, 0, nmax}], x]]
Table[Sum[Sum[PartitionsP[d - k] PartitionsQ[k], {k, 0, d}], {d, Divisors[n]}], {n, 43}]

Formula

G.f.: Sum_{k>=1} A015128(k)*x^k/(1 - x^k).

G.f.: Sum_{k>=1} (1/theta_4(x^k) - 1), where theta_4() is the Jacobi theta function.

a(n) = Sum_{d|n} A015128(d).