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

%I A176085 #51 Jan 26 2025 19:20:51
%S A176085 0,1,3,11,41,155,591,2267,8735,33775,130965,509015,1982269,7732659,
%T A176085 30208749,118167055,462760369,1814091011,7118044023,27952660883,
%U A176085 109853552255,432021606103,1700093447847,6694137523051,26372544576331,103950885100775,409928481296331
%N A176085 a(n) = A136431(n,n).
%C A176085 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
%H A176085 Vincenzo Librandi, <a href="/A176085/b176085.txt">Table of n, a(n) for n = 0..300</a>
%F A176085 a(n+1) - 4*a(n) = -A081696(n-1).
%F A176085 From _Vaclav Kotesovec_, Oct 21 2012: (Start)
%F A176085 G.f.: x*(x-sqrt(1-4*x))/(sqrt(1-4*x)*(x^2+4*x-1)).
%F A176085 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).
%F A176085 a(n) ~ 4^n/sqrt(Pi*n). (End)
%F A176085 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
%F A176085 Simpler g.f.: x/sqrt(1-4*x)/(x+sqrt(1-4*x)). - _Don Knuth_, May 11 2016
%F A176085 a(n) = A000045(3*n) - A054441(n). - _Hrishikesh Venkataraman_, May 27 2021
%F A176085 a(n) = 4*a(n-1) + a(n-2) - binomial(2*n-4,n-2) for n>=2. - _Hrishikesh Venkataraman_, Jul 02 2021
%F A176085 a(n) = A108617(2n,n)/2. - _Alois P. Heinz_, Jan 26 2025
%p A176085 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
%p A176085 1/(sqrt(1 - 4*x) + 1/x - 4): series(%, x, 27):
%p A176085 seq(coeff(%, x, k), k=0..26); # _Peter Luschny_, May 29 2021
%t A176085 t[n_, k_]:= CoefficientList[ Series[x/(1-x-x^2)/(1-x)^k, {x,0,k}], x][[k+1]]; Array[ t[#, #] &, 20]
%t A176085 Table[Sum[Binomial[2*n-k-1, n-k]*Fibonacci[k], {k,0,n}], {n,0,30}] (* _G. C. Greubel_, Nov 28 2019 *)
%o A176085 (Maxima)
%o A176085 a(n):=sum(fib(k)*binomial(2*n-k-1,n-k),k,1,n); /*  _Vladimir Kruchinin_, Mar 17 2016 */
%o A176085 (PARI) a(n) = sum(k=1, n, fibonacci(k)*binomial(2*n-k-1, n-k)) \\ _Michel Marcus_, Mar 17 2016
%o A176085 (Magma) [(&+[Binomial(2*n-k-1, n-k)*Fibonacci(k): k in [0..n]]): n in [0..30]]; // _G. C. Greubel_, Nov 28 2019
%o A176085 (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
%o A176085 (GAP) List([0..30], n-> Sum([0..n], k-> Binomial(2*n-k-1, n-k)*Fibonacci(k) )); # _G. C. Greubel_, Nov 28 2019
%Y A176085 Cf. A000045, A054441, A108617.
%K A176085 nonn,easy
%O A176085 0,3
%A A176085 _Paul Curtz_, Apr 08 2010