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 A084773 Coefficients of 1/sqrt(1-12*x+4*x^2); also, a(n) is the central coefficient of (1+6x+8x^2)^n. 12
 1, 6, 52, 504, 5136, 53856, 575296, 6225792, 68026624, 748832256, 8291791872, 92255680512, 1030537089024, 11550176206848, 129824329777152, 1462841567576064, 16518691986407424, 186887008999047168, 2117944490818011136, 24038305911245635584, 273199990096465494016 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Number of Delannoy paths from (0,0) to (n,n) with steps U(0,1), H(1,0) and D(1,1) where H and D can choose from two colors each (or where one step is monochrome and the other two are bicolored). - Paul Barry, May 30 2005 2^n*P_n(3), where P_n is the n-th Legendre polynomial. 2^n*LegendreP(n,k) yields the central coefficients of (1+2kx+(k^2-1)x^2)^n, with g.f. 1/sqrt(1-4kx+4x^2) and e.g.f. exp(2kx)BesselI(0,2*sqrt(k^2-1)x). - Paul Barry, May 30 2005 Diagonal of rational functions 1/(1 - x - 2*y - 2*x*y), 1/(1 - x - 2*y*z - 2*x*y*z), 1/(1 - 2*x - y*z - 2*x*y*z). - Gheorghe Coserea, Jul 07 2018 LINKS Harvey P. Dale, Table of n, a(n) for n = 0..939 Hacène Belbachir, Abdelghani Mehdaoui, László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq., Vol. 22 (2019), Article 19.3.5. FORMULA a(n) = 2*A052141(n) = 2^n * A001850(n), n>0. E.g.f.: exp(6x)*Bessel_I(0, 2*sqrt(8)*x); a(n) = Sum_{k=0..floor(n/2)} C(n, k)*C(2(n-k), n)*(-1)^k*3^(n-2k). - Paul Barry, May 30 2005 D-finite with recurrence: n*a(n) + 6*(-2*n+1)*a(n-1) + 4*(n-1)*a(n-2) = 0. - R. J. Mathar, Nov 30 2012 G.f.: G(0)/2, where G(k) = 1 + 1/( 1 - x*(6-2*x)*(2*k+1)/(x*(6-2*x)*(2*k+1) + (k+1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 17 2013 a(n) = 2^n*hyper2F1(-n, -n, 1, 2). - Peter Luschny, May 20 2015 a(n) = A059474(n,n). - Alois P. Heinz, Oct 05 2017 a(n) ~ 2^(n - 5/4) * (1 + sqrt(2))^(2*n + 1) / sqrt(Pi*n). - Vaclav Kotesovec, Jul 11 2018 EXAMPLE G.f.: 1/sqrt(1-2*b*x+(b^2-4*c)*x^2) yields central coefficients of (1+b*x+c*x^2)^n. MAPLE a := n -> 2^n*hypergeom([-n, -n], [1], 2): seq(simplify(a(n)), n=0..20); # Peter Luschny, May 20 2015 MATHEMATICA CoefficientList[Series[1/Sqrt[(1-12x+4x^2)], {x, 0, 20}], x] (* Harvey P. Dale, Dec 13 2017 *) PROG (PARI) for(n=0, 30, t=polcoeff((1+6*x+8*x^2)^n, n, x); print1(t", ")) CROSSREFS Cf. A052141, A001850. See A152254 for another interpretation. Cf. A059474. Sequence in context: A334978 A005948 A027258 * A232821 A127133 A243249 Adjacent sequences:  A084770 A084771 A084772 * A084774 A084775 A084776 KEYWORD nonn AUTHOR Paul D. Hanna, Jun 10 2003 STATUS approved

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Last modified July 23 12:19 EDT 2021. Contains 346259 sequences. (Running on oeis4.)