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a(n) = f(n,n) where f(0,n) = f(n,0) = Fibonacci(n) and f(m,n) = f(m-1,n) + f(m,n-1) + f(m-1,n-1).
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%I #34 Jun 30 2021 09:01:30

%S 0,2,10,52,278,1510,8288,45834,254922,1424252,7986550,44921582,

%T 253320352,1431678194,8106897418,45982821860,261206625526,

%U 1485765938390,8461264982176,48237937154554,275275548126890,1572297656021292,8987888015996790,51417128080562142

%N a(n) = f(n,n) where f(0,n) = f(n,0) = Fibonacci(n) and f(m,n) = f(m-1,n) + f(m,n-1) + f(m-1,n-1).

%C a(n+1)/a(n) tends to A156035.

%F a(n) = 2*Sum_{k=1..n} Fibonacci(k)*(A008288(n-1,n-k) + A008288(n-1,n-k-1)). - _Andrew Howroyd_, May 29 2021

%F G.f.: x*(3*x^2-18*x+3-(x+1)*sqrt(x^2-6*x+1))/((x^2-7*x+1)*(x^2-6*x+1)). - _Alois P. Heinz_, May 29 2021

%F a(n) = ((79-97*n+26*n^2)*a(n-1) + (-9+9*n-2*n^2)*a(n-4) + (107-111*n+26*n^2)*a(n-3) + (-322+352*n-88*n^2)*a(n-2)) / (5-7*n+2*n^2) for n >= 4. - _José María Grau Ribas_, Jun 19 2021

%t F[0, 0] = 0; F[m_, 0] := Fibonacci[m]; F[0, n_] := Fibonacci[n];

%t F[m_, n_] := F[m, n] = F[m - 1 , n ] + F[m , n - 1] + F[m - 1, n - 1];

%t Table[F[n, n], {n, 0, 100}]

%o (PARI) \\ here D(n,k) is A008288(n,k).

%o D(n, k) = {sum(d = 0, min(n,k), binomial(k, d)*binomial(n+k-d, k))}

%o a(n) = {2*sum(k=1, n, fibonacci(k)*(D(n-1,n-k) + D(n-1,n-k-1)))} \\ _Andrew Howroyd_, May 29 2021

%Y Cf. A000045, A001850, A008288, A156035.

%K nonn

%O 0,2

%A _José María Grau Ribas_, May 24 2021