A302197 - OEIS

%I #28 Jun 27 2023 02:34:09

%S 0,1,1,1,0,-4,-10,15,210,504,-3528,-34440,-36960,1512720,11763180,

%T -24549525,-1118467350,-6466860400,62185563440,1297024576848,

%U 3903558763104,-149417396724960,-2150022118411440,3233834859735480,449839942314082320

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

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

%H V. E. Adler and A. B. Shabat, <a href="https://arxiv.org/abs/1810.13198">Volterra chain and Catalan numbers</a>, arXiv:1810.13198 [nlin.SI], 2018.

%H Xing Gao and William F. Keigher, <a href="https://doi.org/10.1080/00927872.2016.1226885">Interlacing of Hurwitz series</a>, Communications in Algebra, 45:5 (2017), 2163-2185, DOI: 10.1080/00927872.2016.1226885 .

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

%F E.g.f. is also the log of e^x times the e.g.f. of A005043. - _Tom Copeland_, Jun 26 2023

%p # first load Maple commands for Hurwitz operations from link in A302189.

%p s:=[seq(binomial(2*n,n)/(n+1),n=0..30)];

%p Hlog(s);

%t 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 *)

%o (Sage)

%o A = PowerSeriesRing(QQ, 'x')

%o f = A([catalan_number(i) for i in range(30)]).ogf_to_egf().log()

%o print(list(f.egf_to_ogf()))

%o # _F. Chapoton_, Apr 11 2020

%Y Cf. A000108, A302189.

%Y Cf. A005043.

%K sign

%O 0,6

%A _N. J. A. Sloane_ and William F. Keigher, Apr 14 2018