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

0,2

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.

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

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 *)

Prog

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

Crossrefs

Sequence in context: A081335 A136783 A227726 * A003645 A081085 A212326

Adjacent sequences: A080606 A080607 A080608 * A080610 A080611 A080612

Keyword

easy,nonn

Author

Emanuele Munarini, Feb 26 2003

Status

approved