A176085 a(n) = A136431(n,n).

0, 1, 3, 11, 41, 155, 591, 2267, 8735, 33775, 130965, 509015, 1982269, 7732659, 30208749, 118167055, 462760369, 1814091011, 7118044023, 27952660883, 109853552255, 432021606103, 1700093447847, 6694137523051, 26372544576331, 103950885100775, 409928481296331

Offset: 0

Author

Paul Curtz, Apr 08 2010

Keywords

• nonn
• easy

Comments

a(n+1) is also the number of sequences of length 2n obeying the regular expression "0^* (1 or 2)^* 3^*" and having sum 3n. For example, a(3)=11 because of the sequences 0033, 0123, 0213, 0222, 1113, 1122, 1212, 1221, 2112, 2121, 2211. - Don Knuth, May 11 2016

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..300

Crossrefs

Cf. A000045, A054441, A108617.

Programs

GAP

List([0..30], n-> Sum([0..n], k-> Binomial(2*n-k-1, n-k)*Fibonacci(k) )); # G. C. Greubel, Nov 28 2019

Magma

[(&+[Binomial(2*n-k-1, n-k)*Fibonacci(k): k in [0..n]]): n in [0..30]]; // G. C. Greubel, Nov 28 2019

Maple

with(combinat); seq( add(binomial(2*n-k-1, n-k)*fibonacci(k), k=0..n), n=0..30); # G. C. Greubel, Nov 28 2019
1/(sqrt(1 - 4*x) + 1/x - 4): series(%, x, 27):
seq(coeff(%, x, k), k=0..26); # Peter Luschny, May 29 2021

Mathematica

t[n_, k_]:= CoefficientList[ Series[x/(1-x-x^2)/(1-x)^k, {x,0,k}], x][[k+1]]; Array[ t[#, #] &, 20]
Table[Sum[Binomial[2*n-k-1, n-k]*Fibonacci[k], {k,0,n}], {n,0,30}] (* G. C. Greubel, Nov 28 2019 *)

Maxima

a(n):=sum(fib(k)*binomial(2*n-k-1,n-k),k,1,n); /*  Vladimir Kruchinin, Mar 17 2016 */

PARI

a(n) = sum(k=1, n, fibonacci(k)*binomial(2*n-k-1, n-k)) \\ Michel Marcus, Mar 17 2016

Sage

[sum(binomial(2*n-k-1, n-k)*fibonacci(k) for k in (0..n)) for n in (0..30)] # G. C. Greubel, Nov 28 2019

Formula

a(n+1) - 4*a(n) = -A081696(n-1).

From Vaclav Kotesovec, Oct 21 2012: (Start)

G.f.: x*(x-sqrt(1-4*x))/(sqrt(1-4*x)*(x^2+4*x-1)).

Recurrence: (n-2)*a(n) = 2*(4*n-9)*a(n-1) - (15*n-38)*a(n-2) - 2*(2*n-5)* a(n-3).

a(n) ~ 4^n/sqrt(Pi*n). (End)

a(n) = Sum_{k=1..n} (F(k)*binomial(2*n-k-1,n-k)), where F(k) = A000045(k). - Vladimir Kruchinin, Mar 17 2016

Simpler g.f.: x/sqrt(1-4*x)/(x+sqrt(1-4*x)). - Don Knuth, May 11 2016

a(n) = A000045(3*n) - A054441(n). - Hrishikesh Venkataraman, May 27 2021

a(n) = 4*a(n-1) + a(n-2) - binomial(2*n-4,n-2) for n>=2. - Hrishikesh Venkataraman, Jul 02 2021

a(n) = A108617(2n,n)/2. - Alois P. Heinz, Jan 26 2025