A302195 Hurwitz inverse of triangular numbers [1,3,6,10,15,...].

1, -3, 12, -64, 441, -3771, 38638, -461742, 6306009, -96885451, 1653938616, -31057949748, 636230845297, -14119481897379, 337448486204586, -8640908986912786, 236015269236658833, -6849355531826261427, 210466462952536609924

Offset: 0

Author

N. J. A. Sloane and William F. Keigher, Apr 14 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..400

Crossrefs

Cf. A000217, A302189.

Programs

Maple

# first load Maple commands for Hurwitz operations from link in A302189.
s:=[seq(n*(n+1)/2,n=1..64)];
Hinv(s);

Mathematica

nmax = 20; CoefficientList[Series[1/(E^x*(1 + 2*x + x^2/2)), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Apr 26 2018 *)

Formula

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

From Vaclav Kotesovec, Apr 26 2018: (Start)

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

a(n) ~ (-1)^n * n! * exp(2 - sqrt(2)) * (1 + 1/sqrt(2))^(n+1) / sqrt(2).

(End)