A263401 -id:A263401 - cp's OEIS frontend

A293254 Expansion of Product_{k>=1} (1 + x^k - x^(k^2)).

1, 0, 1, 1, 0, 2, 2, 2, 2, 3, 4, 4, 6, 7, 8, 9, 9, 14, 15, 17, 22, 23, 27, 33, 36, 43, 48, 56, 64, 72, 84, 94, 109, 124, 136, 157, 178, 197, 225, 253, 286, 319, 358, 401, 451, 500, 559, 628, 694, 774, 863, 958, 1066, 1181, 1310, 1449, 1605, 1778, 1957, 2170

Offset: 0

Vaclav Kotesovec, Oct 04 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A033461, A263401, A276516, A293253.

nmax = 100; CoefficientList[Series[Product[(1+x^k-x^(k^2)), {k, 1, nmax}], {x, 0, nmax}], x]

A319668 Expansion of Product_{k>=1} (1 - x^k - x^(2*k)).

Original entry on oeis.org

1, -1, -2, 0, 0, 3, 1, 3, 1, -2, 0, -3, -6, -4, 1, -8, 1, 2, 5, 5, 4, 9, 13, 7, 3, 1, 3, 7, -16, -9, -17, -13, -21, -5, -25, -33, -3, -3, -9, 22, -6, 11, 29, 29, 57, 37, 40, 31, 58, 18, 35, 40, 37, -24, -36, -34, -29, -60, -54, -98, -74, -124, -113, -156, -71, -35, -140, -46, -16, -61, -25

Offset: 0

Ilya Gutkovskiy, Sep 25 2018

sign

Cf. A000010, A000358, A000726, A109389, A137569, A162891, A263401.

a:=series(mul((1-x^k-x^(2*k)),k=1..100),x=0,71): seq(coeff(a,x,n),n=0..70); # Paolo P. Lava, Apr 02 2019

nmax = 70; CoefficientList[Series[Product[(1 - x^k - x^(2 k)), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 70; CoefficientList[Series[Exp[Sum[Sum[EulerPhi[j] Log[1 - x^(j k) (1 + x^(j k))]/(j k), {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, -Sum[Sum[Sum[EulerPhi[d/j] (Fibonacci[j - 1] + Fibonacci[j + 1]), {j, Divisors[d]}], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 70}]

G.f.: exp(Sum_{k>=1} Sum_{j>=1} phi(j)*log(1 - x^(j*k)*(1 + x^(j*k)))/(j*k)), where phi = Euler totient function (A000010).

cp's OEIS Frontend

A293254 Expansion of Product_{k>=1} (1 + x^k - x^(k^2)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica