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 A092566 Main diagonal of triangle A092565, in which the n-th row polynomial equals the numerator of the n-th convergent of the continued fraction [1 + x + x^2; 1 + x + x^2, 1 + x + x^2, ...]. 35
 1, 1, 3, 7, 22, 63, 191, 573, 1752, 5372, 16597, 51465, 160258, 500551, 1567881, 4922687, 15488481, 48821964, 154147654, 487412324, 1543231353, 4891986889, 15524303265, 49314008259, 156791992914, 498931763064, 1588891019625 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS T(n,k) is the number of lattice paths from (0,0) to (n,k) using steps (1,0), (2,0), (1,1), and (1,2). - Joerg Arndt, Jun 30 2011 Diagonal of rational function 1/(1 - (x + x^2 + x*y + x*y^2)). - Gheorghe Coserea, Aug 06 2018 LINKS Alois P. Heinz, Table of n, a(n) for n = 0..1959 FORMULA a(n) = sum(k=0..n, A037027(n, k)*C(k, n-k) ). O.g.f. A(x) satisfies the equation (27*x^4 - 14*x^3 + 9*x^2 + 14*x - 5)*A(x)^3 + (4-3*x)*A(x) + 1 = 0. - Mark van Hoeij, Apr 16 2013 MAPLE series(RootOf((27*x^4-14*x^3+9*x^2+14*x-5)*y^3+(4-3*x)*y+1, y), x=0, 30); # Mark van Hoeij, Apr 16 2013 MATHEMATICA A037027[n_, k_] := Sum[Binomial[k+j, k]*Binomial[j, n-j-k], {j, 0, n-k}]; A037027[n_, 0] = Fibonacci[n+1]; a[n_] := Sum[A037027[n, k]*Binomial[k, n-k], {k, 0, n}]; Table[a(n), {n, 0, 26}] (* Jean-François Alcover, Jul 18 2011 *) a[0, 0] = 1; a[n_, k_] /; n >= 0 && k >= 0 := a[n, k] = a[n, k-1] + a[n, k-2] + a[n-1, k-1] + a[n-2, k-1]; a[_, _] = 0; a[n_] := a[n, n]; a /@ Range[0, 30] (* Jean-François Alcover, Oct 06 2019, after Joerg Arndt *) PROG (PARI) a(n)=if(n<0, 0, polcoeff(contfracpnqn(vector(n, i, 1+x+x^2))[1, 1], n, x)) (PARI) A037027(n, k)=if(n

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Last modified October 15 09:22 EDT 2019. Contains 328026 sequences. (Running on oeis4.)