A302197 Hurwitz logarithm of Catalan numbers [1,1,2,5,14,...].

0, 1, 1, 1, 0, -4, -10, 15, 210, 504, -3528, -34440, -36960, 1512720, 11763180, -24549525, -1118467350, -6466860400, 62185563440, 1297024576848, 3903558763104, -149417396724960, -2150022118411440, 3233834859735480, 449839942314082320

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.

Links

• V. E. Adler and A. B. Shabat, Volterra chain and Catalan numbers, arXiv:1810.13198 [nlin.SI], 2018.
• Xing Gao and William F. Keigher, Interlacing of Hurwitz series, Communications in Algebra, 45:5 (2017), 2163-2185, DOI: 10.1080/00927872.2016.1226885 .

Crossrefs

Cf. A000108, A302189.

Cf. A005043.

Programs

Maple

# first load Maple commands for Hurwitz operations from link in A302189.
s:=[seq(binomial(2*n,n)/(n+1),n=0..30)];
Hlog(s);

Mathematica

nmax = 30; CoefficientList[Series[2*x + Log[BesselI[0, 2*x] - BesselI[1, 2*x]], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jun 26 2023 *)

Sage

A = PowerSeriesRing(QQ, 'x')
f = A([catalan_number(i) for i in range(30)]).ogf_to_egf().log()
print(list(f.egf_to_ogf()))
# F. Chapoton, Apr 11 2020

Formula

E.g.f. is log of e.g.f. for Catalan numbers.

E.g.f. is also the log of e^x times the e.g.f. of A005043. - Tom Copeland, Jun 26 2023