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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
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
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
KEYWORD
nonn,easy
AUTHOR
Paul D. Hanna, Jun 10 2003, Jun 11 2003
STATUS
approved