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

0,2

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

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Vincent Pilaud, V Pons, Permutrees, arXiv preprint arXiv:1606.09643, 2016

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)

Example

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

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 *)

Prog

(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 */

Crossrefs

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

Sequence in context: A089387 A206902 A275752 * A350645 A330793 A352862

Adjacent sequences: A084865 A084866 A084867 * A084869 A084870 A084871

Keyword

nonn,easy

Author

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

Status

approved