A375444 Expansion of g.f. A(x) satisfying A(x)^2 = A( x^2/(1-2*x)^4 )/(1-2*x).

1, 1, 2, 7, 30, 130, 561, 2460, 11115, 51948, 250551, 1240828, 6274580, 32231322, 167460901, 876998437, 4617448333, 24395086617, 129162020323, 684753458054, 3633159683023, 19287528099428, 102441443882448, 544372928359375, 2894576197980724, 15402989792369740, 82040643327234351

Offset

0,3

Comments

Compare to M(x)^2 = M( x^2/(1-2*x) )/(1-2*x), where M(x) = 1 + x*M(x) + x^2*M(x)^2 is the g.f. of the Motzkin numbers (A001006).

Compare to C(x)^2 = C( x^2/(1-2*x)^2 )/(1-2*x), where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).

Links

Paul D. Hanna, Table of n, a(n) for n = 0..400

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.

(1) A(x)^2 = A( x^2/(1-2*x)^4 )/(1-2*x).

(2) A(x)^4 = A( x^4*y^4 )*y where y = (1-2*x)^2/((1-2*x)^4 - 2*x^2).

(3) A(x^2 + 4*x^3 + 4*x^4) = A( x/(1+2*x) )^2 / (1+2*x).

The radius of convergence r satisfies r = (1 - 2*r)^4, where A(r) = 1/(1-2*r) and r = 0.17610056436947880725475085178711534652...

Example

G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 30*x^4 + 130*x^5 + 561*x^6 + 2460*x^7 + 11115*x^8 + 51948*x^9 + 250551*x^10 + ...
where A(x)^2 = A( x^2/(1-2*x)^4 )/(1-2*x).
RELATED SERIES.
A(x)^2 = 1 + 2*x + 5*x^2 + 18*x^3 + 78*x^4 + 348*x^5 + 1551*x^6 + 6982*x^7 + 32114*x^8 + 151620*x^9 + 734458*x^10 + ...
A(x)^4 = 1 + 4*x + 14*x^2 + 56*x^3 + 253*x^4 + 1188*x^5 + 5598*x^6 + 26456*x^7 + 126278*x^8 + ... + A375454(n+1)*x^n + ...
SPECIFIC VALUES.
Given the radius of convergence r = 0.17610056436947880725475...,
A(r) = 1.5436890126920763615708559718017479865252032976509...
where r = (1-2*r)^4 and A(r) = 1/(1-2*r).
A(1/6) = 1.35888986768048814311476385141914227984504826245...
where A(1/6)^2 = (3/2)*A(9/64).
A(1/7) = 1.23858760007712401376241920277473621006326963714...
where A(1/7)^2 = (7/5)*A(49/625).
A(1/8) = 1.18621527667665867031082807873688257681814274612...
where A(1/8)^2 = (4/3)*A(4/81).
A(1/9) = 1.15430486498931766438966249826580193821574473318...
where A(1/9)^2 = (9/7)*A(81/2401).
A(1/10) = 1.1323205915354275720071052412999606676975412945...
where A(1/10)^2 = (5/4)*A(25/1024).

Prog

(PARI) {a(n) = my(A=[1], Ax=x); for(i=1, n, A = concat(A, 0); Ax=Ser(A);
A[#A] = (1/2)*polcoeff( subst(Ax, x, x^2/(1-2*x)^4 )/(1-2*x) - Ax^2, #A-1) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A001006, A000108, A375454, A375443, A375445.

Sequence in context: A074416 A325577 A097924 * A027136 A353288 A299296

Adjacent sequences: A375441 A375442 A375443 * A375445 A375446 A375447

Keyword

nonn

Author

Paul D. Hanna, Aug 19 2024

Status

approved