A096045 a(n) = B(2*n, 2)/B(2*n) (see formula section).

1, 10, 46, 190, 766, 3070, 12286, 49150, 196606, 786430, 3145726, 12582910, 50331646, 201326590, 805306366, 3221225470, 12884901886, 51539607550, 206158430206, 824633720830, 3298534883326, 13194139533310, 52776558133246

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (5,-4).

Formula

a(n) = B(2*n, 2)/B(2*n), where B(n, p) = Sum_{i=0..n} p^i * Sum_{j=0..i} binomial(n,j)*B(j))) with B(k) = k-th Bernoulli number.

a(n) = 3*4^n - 2.

a(n) = 5*a(n-1) - 4*a(n-2).

a(n) = 4*a(n-1) + 6. First differences give A002063. - Paul Curtz, Jul 07 2008

From G. C. Greubel, Jan 22 2023: (Start)

a(n) = 3*A000302(n) - 2.

G.f.: (1+5*x)/(1-x)*(1-4*x)).

E.g.f.: 3*exp(4*x) - 2*exp(x). (End)

Mathematica

a[n_]:= Sum[2^k*Sum[Binomial[2*n, j]*BernoulliB[j], {j, 0, k}], {k, 0, 2*n}]/BernoulliB[2*n]; Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Jan 14 2015 *)
NestList[4#+6&, 1, 30] (* Harvey P. Dale, Dec 27 2016 *)

Prog

(PARI) a(n)=sum(i=0, 2*n, 2^i*sum(j=0, i, binomial(2*n, j)*bernfrac(j)))/bernfrac(2*n)
(Magma) [3*4^n-2: n in [0..30]]; // Vincenzo Librandi, Aug 13 2011
(Maxima) A096045(n):=3*4^n-2$ makelist(A096045(n), n, 0, 30); /* Martin Ettl, Nov 13 2012 */
(SageMath) [3*4^n-2 for n in range(41)] # G. C. Greubel, Jan 22 2023

Crossrefs

Cf. A000302, A002063, A096046, A096047, A096048.

Sequence in context: A103501 A219003 A003197 * A287090 A183133 A115712

Adjacent sequences: A096042 A096043 A096044 * A096046 A096047 A096048

Keyword

nonn,easy

Author

Benoit Cloitre, Jun 17 2004

Status

approved