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A320941 Expansion of Sum_{k>=1} x^k*(1 + x^k)/(1 - x^k)^4. 1
1, 6, 15, 36, 56, 111, 141, 240, 300, 446, 507, 791, 820, 1161, 1310, 1736, 1786, 2505, 2471, 3346, 3466, 4307, 4325, 5895, 5581, 7026, 7230, 8905, 8556, 11246, 10417, 13176, 13050, 15476, 15106, 19391, 17576, 21495, 21374, 25690, 23822, 30162, 27435, 33707, 32990, 37841, 35721 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Inverse Möbius transform of square pyramidal numbers (A000330).

LINKS

Table of n, a(n) for n=1..47.

N. J. A. Sloane, Transforms

FORMULA

G.f.: Sum_{k>=1} A000330(k)*x^k/(1 - x^k).

a(n) = Sum_{d|n} d*(d + 1)*(2*d + 1)/6.

a(n) = (A000203(n) + 3*A001157(n) + 2*A001158(n))/6.

MAPLE

a:=series(add(x^k*(1+x^k)/(1-x^k)^4, k=1..100), x=0, 48): seq(coeff(a, x, n), n=1..47); # Paolo P. Lava, Apr 02 2019

MATHEMATICA

nmax = 47; Rest[CoefficientList[Series[Sum[x^k (1 + x^k)/(1 - x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x]]

Table[Sum[d (d + 1) (2 d + 1)/6, {d, Divisors[n]}], {n, 47}]

Table[(DivisorSigma[1, n] + 3 DivisorSigma[2, n] + 2 DivisorSigma[3, n])/6, {n, 47}]

CROSSREFS

Cf. A000203, A000330, A001157, A001158, A007437, A059358.

Sequence in context: A103106 A272788 A273151 * A273390 A273451 A272928

Adjacent sequences:  A320938 A320939 A320940 * A320942 A320943 A320944

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Oct 24 2018

STATUS

approved

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Last modified April 12 21:13 EDT 2021. Contains 342932 sequences. (Running on oeis4.)