A296601 L.g.f.: -log(Product_{k>=1} (1 - k*x^k)^k) = Sum_{n>=1} a(n)*x^n/n.

1, 9, 28, 81, 126, 330, 344, 833, 973, 1754, 1332, 5034, 2198, 5658, 8688, 13313, 4914, 28779, 6860, 54106, 45752, 33482, 12168, 254954, 93751, 78906, 255880, 505698, 24390, 1510700, 29792, 1671169, 1791312, 647114, 2819544, 12637371, 50654, 2282346, 14779520, 34058298, 68922, 68084220

Offset

1,2

Links

Seiichi Manyama, Table of n, a(n) for n = 1..5000

Index entries for sequences related to sums of divisors

Formula

G.f.: Sum_{k>=1} k^3*x^k/(1 - k*x^k).

a(n) = Sum_{d|n} d^(n/d+2).

a(p) = p^3 + 1 where p is a prime.

From Seiichi Manyama, Jun 24 2019: (Start)

Suppose given two sequences f(n) and g(n), n>0, we define a new sequence a(n), n>0, by a(n) = Sum_{d|n} d*f(d)*g(d)^(n/d).

L.g.f.: -log(Product_{n>0} (1 - g(n)*x^n)^f(n)) = Sum_{n>0} a(n)*x^n/n. (See A266964.)

If we set f(n) = n and g(n) = n, we get this sequence. (End)

Example

L.g.f.: L(x) = x + 9*x^2/2 + 28*x^3/3 + 81*x^4/4 + 126*x^5/5 + 330*x^6/6 + 344*x^7/7 + 833*x^8/8 + 973*x^9/9 + ...
exp(L(x)) = 1 + x + 5*x^2 + 14*x^3 + 42*x^4 + 103*x^5 + 289*x^6 + 690*x^7 + 1771*x^8 + 4206*x^9 + ... + A266941(n)*x^n + ...

Mathematica

nmax = 42; Rest[CoefficientList[Series[-Log[Product[(1 - k x^k)^k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]
nmax = 42; Rest[CoefficientList[Series[Sum[k^3 x^k/(1 - k x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
a[n_] := Sum[d^(n/d + 2), {d, Divisors[n]}]; Table[a[n], {n, 1, 42}]

Prog

(PARI) N=66; x='x+O('x^N); Vec(x*deriv(-log(prod(k=1, N, (1-k*x^k)^k)))) \\ Seiichi Manyama, Jun 02 2019

Crossrefs

Column k=2 of A308502.

Cf. A001157, A078308, A266941, A266964.

Sequence in context: A277065 A001158 A171215 * A294567 A053819 A294287

Adjacent sequences: A296598 A296599 A296600 * A296602 A296603 A296604

Keyword

nonn

Author

Ilya Gutkovskiy, May 20 2018

Status

approved