A305620 Expansion of e.g.f. log(1 + Sum_{k>=1} x^(k^2)/k^2).

1, -1, 2, 0, -6, 60, -540, 3780, 12600, -199080, 3074400, -45738000, 511434000, -5621616000, 55394539200, 960323364000, -24001273296000, 498178528848000, -9994137465312000, 156104172544320000, -2076607873660320000, 18061446353670720000, 206725394268993600000

Offset

1,3

Links

Robert Israel, Table of n, a(n) for n = 1..450

Example

E.g.f.: A(x) = x - x^2/2! + 2*x^3/3! - 6*x^5/5! + 60*x^6/6! - 540*x^7/7! + ...
exp(A(x)) = 1 + x + x^4/4 + x^9/9 + x^16/16 + ... + x^A000290(k)/A000290(k) + ...
exp(exp(A(x))-1) = 1 + x + x^2/2! + x^3/3! + 7*x^4/4! + 31*x^5/5! + ... + A205801(k)*x^k/k! + ... = Product_{j>=1} 1/(1 - x^j)^(A008836(j)/j).

Maple

N:= 50: # for a(1)..a(N)
g:= log(1 + add(x^(k^2)/k^2, k=1..floor(sqrt(N)))):
S:= series(g, x, N+1):
seq(coeff(S, x, n)*n!, n=1..N); # Robert Israel, Jun 07 2018

Mathematica

nmax = 23; Rest[CoefficientList[Series[Log[1 + Sum[x^k^2/k^2, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!]
nmax = 23; Rest[CoefficientList[Series[Log[1 + Log[Product[1/(1 - x^k)^(LiouvilleLambda[k]/k), {k, 1, nmax}]]], {x, 0, nmax}], x] Range[0, nmax]!]
a[n_] := a[n] = Boole[IntegerQ[n^(1/2)]] (n - 1)! - Sum[k Binomial[n, k] Boole[IntegerQ[(n - k)^(1/2)]] (n - k - 1)! a[k], {k, 1, n - 1}]/n; a[0] = 0; Table[a[n], {n, 23}]

Crossrefs

Cf. A000290, A008836, A162552, A205801, A205804.

Sequence in context: A079203 A151751 A196354 * A294470 A283654 A291373

Adjacent sequences: A305617 A305618 A305619 * A305621 A305622 A305623

Keyword

sign

Author

Ilya Gutkovskiy, Jun 06 2018

Status

approved