A302189 Hurwitz inverse of squares [1,4,9,16,...].

1, -4, 23, -184, 1933, -25316, 397699, -7288408, 152650649, -3596802148, 94165506031, -2711813462744, 85195437862693, -2899579176456964, 106276755720182363, -4173542380352243896, 174823612884063939889, -7780800729631450594628

Offset: 0

Author

N. J. A. Sloane and William F. Keigher, Apr 12 2018

Keywords

• sign

Comments

In the ring of Hurwitz sequences all members have offset 0.

References

• Xing Gao and William F. Keigher, Interlacing of Hurwitz series, Communications in Algebra, 45:5 (2017), 2163-2185, DOI: 10.1080/00927872.2016.1226885

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..387
• N. J. A. Sloane, Maple programs for operations on Hurwitz sequences

Crossrefs

Cf. A302870.

Programs

Maple

# first load Maple commands for Hurwitz operations from link
s:=[seq(n^2,n=1..64)];
Hinv(s);

Mathematica

nmax = 20; CoefficientList[Series[1/(E^x*(1 + 3*x + x^2)), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Apr 15 2018 *)

Formula

E.g.f. = 1 / Sum_{n >= 0} (n+1)^2*x^n/n!.

From Vaclav Kotesovec, Apr 15 2018: (Start)

E.g.f: exp(-x)/(1 + 3*x + x^2).

a(n) ~ (-1)^n * n! * exp(1/phi^2) * phi^(2*n + 2) / sqrt(5), where phi = A001622 = (1 + sqrt(5))/2 is the golden ratio.

(End)