A294957 -id:A294957 - cp's OEIS frontend

A294956 a(n) = Sum_{d|n} d^(d + n/d).

1, 9, 82, 1041, 15626, 280212, 5764802, 134221889, 3486785131, 100000078254, 3138428376722, 106993207077516, 3937376385699290, 155568095598166344, 6568408355713287812, 295147905180426634241, 14063084452067724991010

Offset: 1

Seiichi Manyama, Nov 12 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..385

Cf. A055225, A062796, A294957.

sd[n_]:=Total[#^(#+n/#)&/@Divisors[n]]; Array[sd,20] (* Harvey P. Dale, Mar 28 2021 *)

PARI
```
a(n) = sumdiv(n, d, d^(d+n/d));
```

N=20; x='x+O('x^N); Vec(x*deriv(-log(prod(k=1, N, (1-k*x^k)^(k^(k-1)))))) \\ Seiichi Manyama, Jun 09 2019

my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, k^(k+1)*x^k/(1-k*x^k))) \\ Seiichi Manyama, Jan 11 2023

L.g.f.: -log(Product_{k>=1} (1 - k*x^k)^(k^(k-1))) = Sum_{k>=1} a(k)*x^k/k. - Seiichi Manyama, Jun 09 2019

G.f.: Sum_{k>0} k^(k+1) * x^k / (1 - k * x^k). - Seiichi Manyama, Jan 11 2023

A295234 Expansion of Product_{k>=1} (1 - k*x^k)^(k^(k-1)).

Original entry on oeis.org

1, -1, -4, -23, -225, -2765, -42291, -758931, -15672042, -365632740, -9512462314, -273071185192, -8574979449941, -292421476560437, -10762598186760785, -425244979326332068, -17953805056325313497, -806668085786772161511

Offset: 0

Seiichi Manyama, Nov 18 2017

sign

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -n^(n-1), g(n) = n.

Seiichi Manyama, Table of n, a(n) for n = 0..386

Cf. A266964, A294956, A294957.

N=66; x='x+O('x^N); Vec(prod(k=1, N, (1-k*x^k)^k^(k-1)))

Convolution inverse of A294957.

a(0) = 1 and a(n) = -(1/n) * Sum_{k=1..n} A294956(k)*a(n-k) for n > 0.

cp's OEIS Frontend

A294956 a(n) = Sum_{d|n} d^(d + n/d).

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A295234 Expansion of Product_{k>=1} (1 - k*x^k)^(k^(k-1)).

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