A156909 G.f.: A(x) = 1 + x*exp( Sum_{k>=1} [A(-(-1)^k*x) - 1]^k/k ).

1, 1, 1, 2, 3, 10, 18, 70, 135, 566, 1134, 4972, 10206, 46098, 96228, 443946, 938223, 4397730, 9382230, 44523232, 95698746, 458639492, 991787004, 4791683932, 10413763542, 50652087010, 110546105292, 540758574440, 1184422556700

Offset: 0

Author

Paul D. Hanna, Mar 04 2009

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 3*x^4 + 10*x^5 + 18*x^6 + 70*x^7 + ...
...
A(x) = 1 + x*exp( [A(x)-1] + [A(-x)-1]^2/2 + [A(x)-1]^3/3 + [A(-x)-1]^4/4 + ...).
		

Crossrefs

Cf. A156907, A156908.

Cf. A157674. - Paul D. Hanna, Mar 05 2009

Programs

PARI

{a(n)=local(A=1+x+x*O(x^n)); for(i=1,n,A=1+x*exp(-sum(k=1,n,(subst(A,x,(-1)^k*x+x*O(x^n))-1)^k/k))); polcoeff(A,n)}

PARI

{a(n)=local(B=(7-sqrt(1-12*x^2+x^2*O(x^n)))/6);polcoeff(B+sqrt(B^2-B),n)} \\ Paul D. Hanna, Mar 05 2009

Formula

From Paul D. Hanna, Mar 05 2009: (Start)

G.f.: A(x) = B(x) + sqrt(12*B(x) - 12 - 3*x^2)/3

where B(x) = (7-sqrt(1-12*x^2))/6 = A(x)*A(-x) = (A(x)+A(-x))/2 = 1 + x^2/(4-3*B(x)).

Lim_{n->infinity} a(2n)/a(2n-1) = 12^(1/3); lim_{n->infinity} a(2n+1)/a(2n) = 12^(2/3). (End)

D-finite with recurrence: 288*(n-6)*(n-5)*(n-4)*(n-3)*a(n-5) + 24*(n-4)*(n-3)*(52*n^2-378*n+761)*a(n-3) + 2*(n-1)*(181*n^3-271*n^2-950*n+1752)*a(n-1) - (n-1)*(n+1)*(87*n^2+38*n+48)*a(n+1) + 4*(n+1)*(n+2)*(n+3)*(n-1)*a(n+3) = 0. - Georg Fischer, Jul 15 2025