A122446 G.f. satisfies: A(x) = 1 + x*A(x)^2 + 2*x^2*(A(x)^2 - A(x)); equals the base sequence of pendular trinomial triangle A122445.

1, 1, 2, 7, 24, 88, 336, 1321, 5316, 21788, 90640, 381750, 1624592, 6975136, 30177056, 131428917, 575765820, 2535433668, 11216757104, 49829385786, 222193501760, 994153952528, 4461915817760, 20082611971226, 90625360612296

Offset: 0

Author

Paul D. Hanna, Sep 07 2006

Keywords

• nonn

Comments

Functional equation for the g.f. is derived from the recurrence of the pendular triangle A122445. Iterated convolutions of this sequence with the central terms (A122447) generates all diagonals of A122445. For example: A122448 = A122446 * A122447; A122449 = A122446^2 * A122447.

Diagonal sums of triangle T with T(n,k) = 2^k*A133336(n,k). - Philippe Deléham, Nov 10 2009

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A122445, A122447, A122448, A122449, A122450, A122451, A122452.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!( (1+2*x^2 -Sqrt(1-4*x-4*x^2+4*x^4))/(2*x*(1+2*x)) )); // G. C. Greubel, Mar 16 2021

Maple

m:=30; S:=series( (1+2*x^2 -sqrt(1-4*x-4*x^2+4*x^4))/(2*x*(1+2*x)), x, m+1):
seq(coeff(S, x, j), j=0..m); # G. C. Greubel, Mar 16 2021

Mathematica

CoefficientList[Series[(1+2*x^2-Sqrt[1-4*x-4*x^2+4*x^4])/(2*x*(1+2*x)), {x, 0, 20}], x] (* Vaclav Kotesovec, Sep 17 2013 *)

PARI

{a(n)=polcoeff(2/(1+2*x^2+sqrt(1-4*x-4*x^2+4*x^4+x*O(x^n))),n)}

Sage

def A122446_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( (1+2*x^2 -sqrt(1-4*x-4*x^2+4*x^4))/(2*x*(1+2*x)) ).list()
A122446_list(30) # G. C. Greubel, Mar 16 2021

Formula

G.f.: A(x) = (1 + 2*x^2 - sqrt(1 -4*x -4*x^2 +4*x^4))/(2*x*(1+2*x)).

Recurrence: (n+1)*a(n) = 2*(n-2)*a(n-1) + 12*(n-1)*a(n-2) + 8*(n-2)*a(n-3) - 4*(n-5)*a(n-4) - 8*(n-5)*a(n-5). - Vaclav Kotesovec, Sep 17 2013

a(n) ~ c*d^n/(sqrt(Pi)*n^(3/2)), where d = (1/6)*(6+sqrt(6*(10 + 2^(2/3)*(43-3*sqrt(177))^(1/3) + 2^(2/3)*(43+3*sqrt(177))^(1/3))) + sqrt(6*(20-2^(2/3)*(43-3*sqrt(177))^(1/3) - 2^(2/3)*(43+3*sqrt(177))^(1/3) + 24*sqrt(6/(10+2^(2/3)*(43-3*sqrt(177))^(1/3) + 2^(2/3)*(43+3*sqrt(177))^(1/3)))))) = 4.797536514160165558... is the root of the equation 4 - 4*d^2 - 4*d^3 + d^4 = 0 and c = 0.908214882020417619380249683... is the positive root of the equation -59 - 944*c^2 - 2032*c^4 - 320*c^6 + 5184*c^8 = 0. - Vaclav Kotesovec, Sep 17 2013, updated Mar 18 2024