A375443 Expansion of g.f. A(x) satisfying A(x)^2 = A( x^2/(1-2*x)^3 )/(1-2*x).

1, 1, 2, 6, 21, 77, 290, 1122, 4462, 18210, 76028, 323524, 1398071, 6115707, 27008516, 120162616, 537702116, 2417043444, 10904533054, 49343555890, 223851302500, 1017798552096, 4637127493554, 21167261603078, 96799606576699, 443460169286639, 2035144213216892, 9355941004378324

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..515

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)^3 )/(1-2*x).

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

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

The radius of convergence r satisfies r = (1 - 2*r)^3, where A(r) = 1/(1-2*r) and r = (1/12)*(6 + (6*sqrt(87) - 54)^(1/3) - (6*sqrt(87) + 54)^(1/3)) = 0.20512274384927080786...

Example

G.f.: A(x) = x + 3*x^2 + 9*x^3 + 31*x^4 + 117*x^5 + 459*x^6 + 1835*x^7 + 7449*x^8 + 30711*x^9 + 128601*x^10 + ...
where A(x)^2 = A( x^2/(1-2*x)^3 )/(1-2*x).
RELATED SERIES.
A(x)^2 = 1 + 2*x + 5*x^2 + 16*x^3 + 58*x^4 + 220*x^5 + 854*x^6 + 3384*x^7 + 13693*x^8 + 56546*x^9 + 237897*x^10 + ...
A(x)^3 = 1 + 3*x + 9*x^2 + 31*x^3 + 117*x^4 + 459*x^5 + 1835*x^6 + 7449*x^7 + 30711*x^8 + ... + A375453(n+1)*x^n + ...
SPECIFIC VALUES.
Given the radius of convergence r = 0.2051227438492708078605991264519...,
A(r) = 1.6956207695598620574163671001175353426181793882085...
  where r = (1-2*r)^3 and A(r) = 1/(1-2*r).
A(1/5) = 1.51884977058839576453094931523796453209831069839...
  where A(1/5)^2 = (5/3)*A(5/27).
A(1/6) = 1.29543251347110009761686143135328534086163706795...
  where A(1/6)^2 = (6/4)*A(6/64).
A(1/7) = 1.22025427535592887335278669533719663766721910803...
  where A(1/7)^2 = (7/5)*A(7/125).
A(1/10) = 1.12934836581956838019397695630366800332615427708...
  where A(1/10)^2 = (10/8)*A(10/512).

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)^3 )/(1-2*x) - Ax^2, #A-1) ); H=Ax; A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A001006, A000108, A375453, A375444, A375445.

Sequence in context: A279561 A360150 A294048 * A063023 A150188 A150189

Adjacent sequences: A375440 A375441 A375442 * A375444 A375445 A375446

Keyword

nonn

Author

Paul D. Hanna, Aug 18 2024

Status

approved