A210672 a(0)=1; thereafter a(n) = 2*Sum_{k=1..n} binomial(2n,2k)*a(n-k).

1, 2, 26, 842, 50906, 4946282, 704888186, 138502957322, 35887046307866, 11855682722913962, 4863821092813045946, 2425978759725443056202, 1445750991051368583278426, 1014551931766896667943384042, 828063237870027116855857421306, 777768202388460616924079724057482

Offset: 0

Author

N. J. A. Sloane, Mar 28 2012

Keywords

• nonn
• easy

Comments

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.

Exp( Sum_{n >= 1} a(n)*x^n/n) is the o.g.f. for A255929. - Peter Bala, Mar 13 2015

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

Links

• G. C. Greubel, Table of n, a(n) for n = 0..220
• Hacène Belbachir, Yahia Djemmada, On central Fubini-like numbers and polynomials, arXiv:1811.06734 [math.CO], 2018. See p. 4.

Crossrefs

Cf. A028296, A094088, A210657, A210674, A210676, A255929, A241171.

Programs

Maple

f:=proc(n,k) option remember;  local i;
if n=0 then 1
else k*add(binomial(2*n,2*i)*f(n-i,k),i=1..floor(n)); fi; end;
g:=k->[seq(f(n,k),n=0..40)];
g(2);

Mathematica

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 *)

Formula

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

E.g.f.: 1/(3-2*cosh(x)) (even coefficients). - Vaclav Kotesovec, Mar 14 2015

a(n) = Sum_{k = 0..2*n} A163626(2*n,k)*A000045(n+1). - Philippe Deléham, May 25 2015

a(n) = Sum_{k=0..n} A241171(n, k)*2^k. - Peter Luschny, Sep 03 2022