cp's OEIS Frontend

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

%I A302197 #28 Jun 27 2023 02:34:09
%S A302197 0,1,1,1,0,-4,-10,15,210,504,-3528,-34440,-36960,1512720,11763180,
%T A302197 -24549525,-1118467350,-6466860400,62185563440,1297024576848,
%U A302197 3903558763104,-149417396724960,-2150022118411440,3233834859735480,449839942314082320
%N A302197 Hurwitz logarithm of Catalan numbers [1,1,2,5,14,...].
%C A302197 In the ring of Hurwitz sequences all members have offset 0.
%H A302197 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 A302197 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 A302197 E.g.f. is log of e.g.f. for Catalan numbers.
%F A302197 E.g.f. is also the log of e^x times the e.g.f. of A005043. - _Tom Copeland_, Jun 26 2023
%p A302197 # first load Maple commands for Hurwitz operations from link in A302189.
%p A302197 s:=[seq(binomial(2*n,n)/(n+1),n=0..30)];
%p A302197 Hlog(s);
%t A302197 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 A302197 (Sage)
%o A302197 A = PowerSeriesRing(QQ, 'x')
%o A302197 f = A([catalan_number(i) for i in range(30)]).ogf_to_egf().log()
%o A302197 print(list(f.egf_to_ogf()))
%o A302197 # _F. Chapoton_, Apr 11 2020
%Y A302197 Cf. A000108, A302189.
%Y A302197 Cf. A005043.
%K A302197 sign
%O A302197 0,6
%A A302197 _N. J. A. Sloane_ and William F. Keigher, Apr 14 2018