A304870 - OEIS

A304870

L.g.f.: log(Product_{k>=1} (1 + x^k)^(k^k)) = Sum_{n>=1} a(n)*x^n/n.

1, 7, 82, 1015, 15626, 279862, 5764802, 134216695, 3486784483, 99999984382, 3138428376722, 106993205100070, 3937376385699290, 155568095552047430, 6568408355712906332, 295147905179218607095, 14063084452067724991010, 708235345355334189853093, 37589973457545958193355602

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OFFSET

1,2

LINKS

Table of n, a(n) for n=1..19.

Index entries for sequences related to sums of divisors

FORMULA

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

a(n) = Sum_{d|n} (-1)^(n/d+1)*d^(d+1).

a(p) = p^(p+1) + 1 where p is an odd prime.

EXAMPLE

L.g.f.: L(x) = x + 7*x^2/2 + 82*x^3/3 + 1015*x^4/4 + 15626*x^5/5 + 279862*x^6/6 + 5764802*x^7/7 + 134216695*x^8/8 + 3486784483*x^9/9 + ...

exp(L(x)) = 1 + x + 4*x^2 + 31*x^3 + 289*x^4 + 3495*x^5 + 51268*x^6 + 891152*x^7 + 17926913*x^8 + 409907600*x^9 + ... + A261053(n)*x^n + ...

MATHEMATICA

nmax = 19; Rest[CoefficientList[Series[Log[Product[(1 + x^k)^k^k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]

nmax = 19; Rest[CoefficientList[Series[Sum[k^(k + 1) x^k/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x]]

a[n_] := Sum[(-1)^(n/d + 1) d^(d + 1), {d, Divisors[n]}]; Table[a[n], {n, 1, 19}]

CROSSREFS

Cf. A007778, A062796, A261053, A283498.

Sequence in context: A285062 A253265 A338684 * A361714 A191804 A243672

Adjacent sequences: A304867 A304868 A304869 * A304871 A304872 A304873

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, May 20 2018

STATUS

approved