A080609 Binomial transform of central Delannoy numbers A001850.

1, 4, 20, 112, 664, 4064, 25376, 160640, 1027168, 6618496, 42904960, 279503360, 1828222720, 11999226880, 78984381440, 521218322432, 3447059138048, 22840932997120, 151607254267904, 1007830488424448, 6708862677274624

Offset: 0

Author

Emanuele Munarini, Feb 26 2003

Keywords

• easy
• nonn

Comments

The Hankel transform (see A001906 for definition) of this sequence is A036442: 1, 4, 32, 512, 16384, ... . - Philippe Deléham, Jul 03 2005

Coefficient of x^n in (1 + 4*x + 2*x^2)^n - N-E. Fahssi, Jan 17 2008

Number of paths from (0,0) to (n,0) using only steps U=(1,1), H=(1,0) and D=(1,-1), U can have 2 colors and H can have 4 colors. - N-E. Fahssi, Jan 27 2008

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Hacène Belbachir, Abdelghani Mehdaoui, László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.
• Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.

Programs

Mathematica

Table[SeriesCoefficient[Series[1/Sqrt[1-8x+8x^2], {x, 0, n}], n], {n, 0, 12}]
Table[LegendreP[n, Sqrt[2]] 8^(n/2), {n, 0, 20}] (* Vladimir Reshetnikov, Nov 01 2015 *)

PARI

x='x+O('x^66); Vec(1/sqrt(1-8*x+8*x^2)) \\ Joerg Arndt, May 07 2013

Formula

G.f.: 1 / sqrt( 1 - 8*x + 8*x^2 ).

a(n) = Sum_{k=0..n} binomial(n,k) * A001850(k).

E.g.f.: exp(4*x)*BesselI(0, 2*sqrt(2)*x). - Vladeta Jovovic, Mar 21 2004

Recurrence: n*a(n) = 4*(2*n-1)*a(n-1) - 8*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 13 2012

a(n) ~ sqrt(1+sqrt(2))*(4+2*sqrt(2))^n/sqrt(2*Pi*n). - Vaclav Kotesovec, Oct 13 2012

G.f.: G(0), where G(k)= 1 + 4*x*(1-x)*(4*k+1)/(2*k+1 - 2*x*(1-x)*(2*k+1)*(4*k+3)/(2*x*(1-x)*(4*k+3) + (k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 22 2013

a(n) = LegendreP_n(sqrt(2))*8^(n/2). - Vladimir Reshetnikov, Nov 01 2015