A084868 Main diagonal of symmetric square table A084867, in which the antidiagonal sums (A006012) form the first row shifted left.

1, 2, 8, 36, 168, 796, 3800, 18216, 87536, 421292, 2029592, 9784088, 47187536, 227651352, 1098523504, 5301727824, 25590307552, 123529362124, 596337248024, 2878947861432, 13899229883024, 67105641925064, 323993230750672

Offset: 0

Author

Paul D. Hanna, Jun 10 2003, Jun 11 2003

Keywords

• nonn
• easy

Comments

The Hankel transform (see A001906 for definition) of this sequence is A000302 (powers of 4): 1, 4, 16, 64, 256, 1024, ... - Philippe Deléham, Aug 17 2005

Examples

			1 + 2*x + 8*x^2 + 36*x^3 + 168*x^4 + 796*x^5 + 3800*x^6 + 18216*x^7 + ...
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Vincent Pilaud, V Pons, Permutrees, arXiv preprint arXiv:1606.09643, 2016

Crossrefs

Cf. A006012, A011782, A028329, A084867, A026671, A081696.

Programs

Maple

1/(1-x/(sqrt(1/4-x))): series(%,x,23): seq(coeff(%,x,n),n=0..22); # Peter Luschny, Feb 06 2017

Mathematica

Table[SeriesCoefficient[((1-4*x)+2*x*Sqrt[1-4*x])/(1-4*x-4*x^2),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 14 2012 *)

PARI

{a(n) = if( n<0, 0, polcoeff((1 - 4*x + 2*x * sqrt(1 - 4*x + x * O(x^n))) /(1 - 4*x - 4*x^2), n))} /* Michael Somos, Jan 05 2012 */

Formula

Differential equation: (16*x^3 + 12*x^2 - 8*x + 1) * x*(d/dx)A(x) + (8x^3 - 12*x^2 + 6*x - 1) * A(x) + (8x^2 - 6*x + 1) = 0.

G.f.: ((1 - 4*x) + 2*x * sqrt(1 - 4*x)) / (1 - 4*x - 4*x^2). a(n) * (n-1) = a(n-1) * (8*n - 14) - a(n-2) * 12*(n-3) - a(n-3) * 8*(2*n - 5), n > 2. Hankel number wall zig-zag diagonal is A011782. - Michael Somos, Sep 14 2003

INVERT transform of A028329 (offset 1). - Michael Somos, Jan 05 2012

G.f.: (1-2*x*f(x))/(1-2*x*f(x)-2*x) where f(x) is the g.f. of A000108 (Catalan numbers). - Philippe Deléham, Jan 30 2012

a(n) ~ (1-1/sqrt(2))*(2+2*sqrt(2))^n. - Vaclav Kotesovec, Oct 14 2012

From Peter Bala, Feb 05 2017: (Start)

G.f: sqrt(1 - 4*x)/(sqrt(1 - 4*x) - 2*x) = 1/(1 - 2*x/(1 - 2*x/(1 - x/(1 - x/(1 - x/(1 - ...)))))) (continued fraction). Cf. A026671, A081696.

Catalan transform of A006012, that is, equals A106566*A006012, as noted by R. J. Mathar. (End)