This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A115125 #33 Aug 06 2024 08:59:44
%S A115125 1,2,4,16,80,448,2688,16896,109824,732160,4978688,34398208,240787456,
%T A115125 1704034304,12171673600,87636049920,635361361920,4634400522240,
%U A115125 33985603829760,250420238745600,1853109766717440,13765958267043840,102618961627054080,767411365211013120
%N A115125 A sequence related to Catalan numbers A000108.
%C A115125 Essentially identical to A025225.
%C A115125 The convolution of this sequence with the sequence {(-1)^n} is A064062 (see also A062992).
%C A115125 The sequence A064062 appears in the Derrida et al. 1992 reference (see A064094) for alpha=2, beta=1 (or alpha=1, beta=2).
%H A115125 G. C. Greubel, <a href="/A115125/b115125.txt">Table of n, a(n) for n = 0..1000</a>
%H A115125 B. Derrida, E. Domany and D. Mukamel, <a href="https://dx.doi.org/10.1007/BF01050430">An exact solution of a one-dimensional asymmetric exclusion model with open boundaries</a>, J. Stat. Phys. 69, 1992, 667-687; eqs. (20), (21), p. 672.
%F A115125 a(n) = C(n-1)*2^n, n>=1, a(0):=1, with C(n):=A000108(n) (Catalan).
%F A115125 G.f.: 1 + (2*x)*c(2*x) with c(x):=(1-sqrt(1-4*x))/(2*x), the o.g.f. of Catalan numbers A000108.
%F A115125 a(n) = A025225(n), n>0. - _R. J. Mathar_, Aug 11 2008
%F A115125 G.f.: (3 - sqrt(1-8*x))/2 = 2 - U(0) where U(k)=1 - 2*x/U(k+1) ; (continued fraction, 1-step). - _Sergei N. Gladkovskii_, Oct 29 2012
%F A115125 G.f.: 2 - 1/Q(0), where Q(k)= 1 + (8*k+2)*x/(k+1 - x*(2*k+2)*(8*k+6)/(2*x*(8*k+6) + (2*k+3)/Q(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 14 2013
%p A115125 a:= n-> `if`(n=0, 1, 2^n*binomial(2*n-2, n-1)/n):
%p A115125 seq(a(n), n=0..25); # _Alois P. Heinz_, Jul 25 2022
%t A115125 a[0] = 1; a[n_] := 2^n*CatalanNumber[n - 1]; Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Jul 09 2013 *)
%o A115125 (PARI) a(n)=if(n==0,1,polcoeff((1-sqrt(1-8*x+x*O(x^n)))/2,n)); \\ _Joerg Arndt_, May 14 2013
%o A115125 (Magma) [1] cat [2^n*Binomial(2*n-2, n-1)/n: n in [1..30]]; // _G. C. Greubel_, May 03 2018
%Y A115125 Cf. A000108, A025225, A062992, A064062, A064094.
%K A115125 nonn,easy
%O A115125 0,2
%A A115125 _Wolfdieter Lang_, Jan 13 2006