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A087626
Expansion of 2/(1-2x+sqrt(1-4x+4x^3)).
2
1, 2, 5, 13, 36, 104, 311, 955, 2995, 9553, 30896, 101082, 333946, 1112496, 3732955, 12605029, 42800317, 146046819, 500555447, 1722402303, 5948047169, 20607691517, 71610355540, 249520257106, 871614139396, 3051737703526
OFFSET
0,2
LINKS
Jean-Luc Baril and José Luis Ramírez, Descent distribution on Catalan words avoiding ordered pairs of Relations, arXiv:2302.12741 [math.CO], 2023.
FORMULA
G.f.: 2/(1-2x+sqrt(1-4x+4x^3)).
G.f. A(x) satisfies 0 = x^2*(1-x)*A(x)^2 - (1-2*x)*A(x) + 1.
First backwards difference is A056010.
(6+4*n)*a(n)+(-6-4*n)*a(n+1)+(-18-4*n)*a(2+n)+(24+5*n)*a(n+3)+(-6-n)*a(n+4)=0. - Robert Israel, Oct 26 2018
HANKEL transform is A006720(n+2). HANKEL transform with 0 prepended is -A051138.
INVERT transform of A157003. - Michael Somos, Mar 28 2020
EXAMPLE
G.f. = 1 + 2*x + 5*x^2 + 13*x^3 + 36*x^4 + 104*x^5 + 311*x^6 + 955*x^7 + ... - Michael Somos, Mar 28 2020
MAPLE
f:= gfun:-rectoproc({(6+4*n)*a(n)+(-6-4*n)*a(n+1)+(-18-4*n)*a(2+n)+(24+5*n)*a(n+3)+(-6-n)*a(n+4), a(0) = 1, a(1) = 2, a(2) = 5, a(3) = 13}, a(n), remember):
map(f, [$0..50]); # Robert Israel, Oct 26 2018
MATHEMATICA
CoefficientList[Series[2/(1-2x+Sqrt[1-4x+4x^3]), {x, 0, 30}], x] (* Harvey P. Dale, Jun 12 2017 *)
PROG
(PARI) {a(n) = polcoeff(2 / (1 - 2*x + sqrt(1 - 4*x + 4*x^3 + x*O(x^n))), n)};
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Sep 16 2003
STATUS
approved