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A084868 Main diagonal of symmetric square table A084867, in which the antidiagonal sums (A006012) form the first row shifted left. 4
1, 2, 8, 36, 168, 796, 3800, 18216, 87536, 421292, 2029592, 9784088, 47187536, 227651352, 1098523504, 5301727824, 25590307552, 123529362124, 596337248024, 2878947861432, 13899229883024, 67105641925064, 323993230750672 (list; graph; refs; listen; history; text; internal format)
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 * A109980 A186338 A190862

Adjacent sequences:  A084865 A084866 A084867 * A084869 A084870 A084871

KEYWORD

nonn,easy

AUTHOR

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

STATUS

approved

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Last modified March 21 12:12 EDT 2019. Contains 321369 sequences. (Running on oeis4.)