%I #36 Sep 03 2022 12:08:02
%S 1,2,26,842,50906,4946282,704888186,138502957322,35887046307866,
%T 11855682722913962,4863821092813045946,2425978759725443056202,
%U 1445750991051368583278426,1014551931766896667943384042,828063237870027116855857421306,777768202388460616924079724057482
%N a(0)=1; thereafter a(n) = 2*Sum_{k=1..n} binomial(2n,2k)*a(n-k).
%C Consider the sequence defined by a(0) = 1; thereafter a(n) = c*Sum_{k = 1..n} binomial(2n,2k)*a(n-k). For c = -3, -2, -1, 1, 2, 3, 4 this is A210676, A210657, A028296, A094088, A210672, A210674, A249939.
%C Exp( Sum_{n >= 1} a(n)*x^n/n) is the o.g.f. for A255929. - _Peter Bala_, Mar 13 2015
%C The Stirling-Bernoulli transform of Fibonacci(n+1) = 1, 1, 2, 3, 5, 8, 13, ... is 1, 0, 2, 0, 26, 0, 842, 0, 50906, 0, ... - _Philippe Deléham_, May 25 2015
%H G. C. Greubel, <a href="/A210672/b210672.txt">Table of n, a(n) for n = 0..220</a>
%H Hacène Belbachir, Yahia Djemmada, <a href="https://arxiv.org/abs/1811.06734">On central Fubini-like numbers and polynomials</a>, arXiv:1811.06734 [math.CO], 2018. See p. 4.
%F a(n) ~ 2*sqrt(Pi/5) * n^(2*n+1/2) / (exp(2*n) * (log((1+sqrt(5))/2))^(2*n+1)). - _Vaclav Kotesovec_, Mar 13 2015
%F E.g.f.: 1/(3-2*cosh(x)) (even coefficients). - _Vaclav Kotesovec_, Mar 14 2015
%F a(n) = Sum_{k = 0..2*n} A163626(2*n,k)*A000045(n+1). - _Philippe Deléham_, May 25 2015
%F a(n) = Sum_{k=0..n} A241171(n, k)*2^k. - _Peter Luschny_, Sep 03 2022
%p f:=proc(n,k) option remember; local i;
%p if n=0 then 1
%p else k*add(binomial(2*n,2*i)*f(n-i,k),i=1..floor(n)); fi; end;
%p g:=k->[seq(f(n,k),n=0..40)];
%p g(2);
%t nmax=20; Table[(CoefficientList[Series[1/(3-2*Cosh[x]), {x, 0, 2*nmax}], x] * Range[0, 2*nmax]!)[[2*n+1]], {n,0,nmax}] (* _Vaclav Kotesovec_, Mar 14 2015 *)
%Y Cf. A028296, A094088, A210657, A210674, A210676, A255929, A241171.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, Mar 28 2012