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, 211106232532990, 844424930131966
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (5,-4).
Programs
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Magma
[3*4^n-2: n in [0..30]]; // Vincenzo Librandi, Aug 13 2011
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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 *) -
Maxima
A096045(n):=3*4^n-2$ makelist(A096045(n),n,0,30); /* Martin Ettl, Nov 13 2012 */
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PARI
a(n)=sum(i=0,2*n,2^i*sum(j=0,i,binomial(2*n,j)*bernfrac(j)))/bernfrac(2*n)
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SageMath
[3*4^n-2 for n in range(41)] # G. C. Greubel, Jan 22 2023
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)