A234854 E.g.f. satisfies: A'(x) = 1 + 4*A(x) + A(x)^2, where A(0)=0.

1, 4, 18, 96, 624, 4896, 45072, 474624, 5619456, 73903104, 1069106688, 16872800256, 288483876864, 5311773904896, 104789944829952, 2205099306123264, 49302137885884416, 1167150882577711104, 29165495002777387008, 767163772371852066816, 21188300138891474632704

Offset

1,2

Links

Table of n, a(n) for n=1..21.

Formula

E.g.f.: Series_Reversion( Integral 1/(1 + 4*x + x^2) dx ).

E.g.f.: (2-sqrt(3))*(exp(2*sqrt(3)*x)-1)/(1 + (4*sqrt(3) - 7)*exp(2*sqrt(3)*x)). - Vaclav Kotesovec, Jan 13 2014

a(n) ~ n! * ((2*sqrt(3))/log(7+4*sqrt(3)))^(n+1). - Vaclav Kotesovec, Jan 13 2014

O.g.f.: x/(1-4*x - 1*2*x^2/(1-8*x - 2*3*x^2/(1-12*x - 3*4*x^2/(1-16*x - 4*5*x^2/(1-20*x - 5*6*x^2/(1- .../(1-4*n*x - n*(n+1)*x^2/(1- ...))))))) (continued fraction). - Paul D. Hanna, Sep 01 2014

Example

E.g.f.: A(x) = x + 4*x^2/2! + 18*x^3/3! + 96*x^4/4! + 624*x^5/5! +...
Related series.
A(x)^2 = 2*x^2/2! + 24*x^3/3! + 240*x^4/4! + 2400*x^5/5! + 25488*x^6/6! +...

Mathematica

Rest[FullSimplify[CoefficientList[Series[(2-Sqrt[3])*(E^(2*Sqrt[3]*x)-1)/(1 + (4*Sqrt[3]-7)*E^(2*Sqrt[3]*x)), {x, 0, 15}], x]*Range[0, 15]!]] (* Vaclav Kotesovec, Jan 13 2014 *)

Prog

(PARI) {a(n)=local(A=x); for(i=1, n, A=intformal(1+4*A+A^2 +x*O(x^n))); n!*polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))
(PARI) {a(n)=local(A=serreverse(intformal(1/(1+4*x+x^2 +x*O(x^n))))); n!*polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))

Crossrefs

Sequence in context: A001563 A094304 A094258 * A334735 A086681 A054139

Adjacent sequences: A234851 A234852 A234853 * A234855 A234856 A234857

Keyword

nonn

Author

Paul D. Hanna, Jan 09 2014

Status

approved