A214724 Expansion of e.g.f.: exp( Sum_{n>=0} x^(n^2+1)/(n^2+1) ).

1, 1, 2, 4, 10, 50, 220, 1240, 6140, 32860, 602200, 5668400, 62030200, 522328600, 4487190800, 62591332000, 715163146000, 30496564010000, 482341877812000, 8342949421288000, 124613700640580000, 1733826182453140000, 36635355834463000000, 597186420007933040000

Offset

0,3

Comments

Conjecture: p | a(n) for n>=p when p is a prime of the form m^2+1 (A002496).

Links

G. C. Greubel, Table of n, a(n) for n = 0..449

Example

E.g.f.: A(x) = 1 + x + 2*x^2/2! + 4*x^3/3! + 10*x^4/4! + 50*x^5/5! + 220*x^6/6! +...
where, by definition,
log(A(x)) = x + x^2/2 + x^5/5 + x^10/10 + x^17/17 + x^26/26 + x^37/37 +...

Mathematica

With[{m=30}, CoefficientList[Series[Exp[Sum[x^(n^2+1)/(1+n^2), {n, 0, m+ 2}]], {x, 0, m}], x]*Range[0, m]!] (* G. C. Greubel, Jan 07 2024 *)

Prog

(PARI) {a(n)=n!*polcoeff(exp(sum(k=0, n, x^(k^2+1)/(k^2+1) + x*O(x^n))), n)}
for(n=0, 21, print1(a(n), ", "))
(Magma)
m:=30;
R<x>:=PowerSeriesRing(Rationals(), m+1);
Coefficients(R!(Laplace( Exp((&+[x^(n^2+1)/(n^2+1): n in [0..m+2]])) ))); // G. C. Greubel, Jan 07 2024
(SageMath)
m=30
def A214724_list(prec):
    P.<x> = PowerSeriesRing(QQ, prec)
    return P( exp(sum(x^(n^2+1)/(n^2+1) for n in range(m+3))) ).egf_to_ogf().list()
A214724_list(m) # G. C. Greubel, Jan 07 2024

Crossrefs

Cf. A002496.

Sequence in context: A000613 A322294 A053500 * A368588 A326325 A080090

Adjacent sequences: A214721 A214722 A214723 * A214725 A214726 A214727

Keyword

nonn

Author

Paul D. Hanna, Jul 26 2012

Status

approved