A259757 G.f. A(x) satisfies A(x)^2 = 1 +x + x*A(x)^5.

1, 1, 2, 8, 35, 169, 862, 4575, 24999, 139700, 794684, 4586377, 26788423, 158054285, 940603900, 5639481930, 34032324940, 206550445064, 1259975808104, 7720835953740, 47504293931640, 293357473042545, 1817649401577760, 11296505623845080, 70402438290940450, 439888817329463279, 2755010697928837222, 17292270772076728414

Offset: 0

Author

Paul D. Hanna, Nov 08 2015

Keywords

• nonn

Comments

Terms appear to equal A011791, apart from offset and an initial 1.

Note that the function G(x) = 1 + x*G(x)^2 (g.f. of A000108) also satisfies this condition: G(x) = 1/G(-x*G(x)^3).

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 8*x^3 + 35*x^4 + 169*x^5 + 862*x^6 + 4575*x^7 + 24999*x^8 + 139700*x^9 + 794684*x^10 +...
where A(x)^2 = 1+x + x*A(x)^5 and
A(x)^2 = 1 + 2*x + 5*x^2 + 20*x^3 + 90*x^4 + 440*x^5 + 2266*x^6 + 12110*x^7 + 66525*x^8 + 373320*x^9 + 2130865*x^10 +...
A(x)^5 = 1 + 5*x + 20*x^2 + 90*x^3 + 440*x^4 + 2266*x^5 + 12110*x^6 + 66525*x^7 + 373320*x^8 + 2130865*x^9 + 12332512*x^10 +...
OTHER RELATIONS.
Let B(x) be defined by B(x*A(x)) = x, then
B(x) =  x - x^2 - 3*x^4 - 3*x^5 - 22*x^6 - 50*x^7 - 240*x^8 - 763*x^9 - 3234*x^10 - 11880*x^11 - 48831*x^12 +...
Let C(x) be defined by C(x*A(x)^2) = A(x), then
C(x) = 1 + x + 3*x^3 - 3*x^4 + 22*x^5 - 50*x^6 + 240*x^7 - 763*x^8 + 3234*x^9 - 11880*x^10 + 48831*x^11 +...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..530

Crossrefs

Cf. A011791, A295537, A295538, A366452.

Cf. A370472, A370473, A370476.

Programs

PARI

{a(n) = my(A=1+x); for(i=1,n, A = sqrt(1+x + x*A^5 +x*O(x^n)) ); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

Formula

G.f. A(x) satisfies [from Paul D. Hanna, Nov 27 2017]:

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

(2) F(A(x)) = x such that F(x) = -(1 - x^2)/(1 + x^5).

(3) A(x) = 1 / A(-x*A(x)^3).

Recurrence: 3*(n-2)*(n-1)*n*(3*n - 1)*(3*n + 1)*a(n) = 6*(n-2)*(n-1)*(2*n - 1)*(3*n - 2)*(3*n - 1)*a(n-1) + 10*(n-2)*(41*n^4 - 164*n^3 + 200*n^2 - 72*n + 3)*a(n-2) + 100*(n-3)*n*(2*n - 3)*(2*n^2 - 6*n + 3)*a(n-3) + 125*(n-4)*(n-3)*(n-1)^2*n*a(n-4). - Vaclav Kotesovec, Nov 18 2017

a(n) ~ 3^(n - 5/2) * 5^n * sqrt((15 + 4*10^(1/3) + 2*10^(2/3))/Pi) / (2*n^(3/2) * (10^(2/3) + 4*10^(1/3) - 11)^(n - 1/2)). - Vaclav Kotesovec, Nov 18 2017

D-finite with recurrence 9*n*(3*n-1)*(3*n+1)*a(n) -6*(3*n-2) *(48*n^2-115*n+83)*a(n-1) +15*(n-1) *(17*n^2-169*n+254)*a(n-2) +50 *(n-3)*(194*n^2-971*n+1200) *a(n-3) +125*(n-4) *(143*n^2-856*n+1265) *a(n-4) +2500*(n-5) *(5*n^2-35*n+59)*a(n-5) +3125*(n-5)*(n-6)*(n-3)*a(n-6)=0. - R. J. Mathar, Nov 16 2023

From Seiichi Manyama, Apr 04 2024: (Start)

G.f. A(x) satisfies A(x) = 1 + x * (1 - A(x) + A(x)^2 - A(x)^3 + A(x)^4).

a(n) = Sum_{k=0..n} binomial(n,k) * binomial(5*k/2+1/2,n)/(5*k+1). (End)