A144010 -id:A144010 - cp's OEIS frontend

A144011 E.g.f. satisfies: A'(x) = 1/(1 - x*A(x))^2 with A(0)=1.

1, 1, 2, 10, 72, 704, 8640, 127968, 2220288, 44179200, 991802880, 24799656960, 683533762560, 20589288993792, 672920058230784, 23717386619136000, 896730039462297600, 36203980633475973120, 1554541449858851143680

Offset: 0

Paul D. Hanna, Sep 10 2008

Vaclav Kotesovec, Table of n, a(n) for n = 0..335
V. Kotesovec, Asymptotic of implicit functions if Fww = 0

Cf. A144010, A238302.

nn=10;Flatten[{1,Table[Subscript[c,j]*j!,{j,1,nn}]/.Solve[Table[SeriesCoefficient[x/(x*(1+Sum[Subscript[c,j]*x^j,{j,1,nn}])-1),{x,0,k}]==SeriesCoefficient[Tan[-Sum[Subscript[c,j]*x^j,{j,1,nn}]],{x,0,k}],{k,0,nn}]]}] (* Vaclav Kotesovec, Jun 15 2013 *)

{a(n)=local(A=1+x); for(i=0, n, A=1+intformal(1/(1-x*A+x*O(x^n))^2 )); n!*polcoeff(A, n)}

E.g.f. A(x) satisfies: A(x) = 1 + Integral 1/(1 - x*A(x))^2 dx.
E.g.f. A(x) satisfies: x/(x*A(x)-1) = tan(1-A(x)). - Vaclav Kotesovec, Jun 15 2013
a(n) ~ GAMMA(1/3) * n^(n-5/6) * (2+Pi)^(n+1/3) / (3^(1/6) * sqrt(Pi) * exp(n) * 2^(n+5/6)). - Vaclav Kotesovec, Feb 23 2014

A173895 E.g.f. satisfies: A'(x) = 1/(1 + x*A(x)) with A(0) = 1.

Original entry on oeis.org

1, 1, -1, 0, 9, -48, 15, 2448, -24927, 23424, 3091311, -47659200, 88056969, 10702667520, -225139993377, 679791291648, 78646340795265, -2128005345251328, 9456106738649631, 1053535684549174272

Offset: 0

Peter Bala, Nov 26 2010

Define a polynomial sequence P_n(x) recursively by
... P_0(x) = 1, and for n >= 1
... P_n(x) = (x-1)*P_(n-1)(x-1)-n*P_(n-1)(x+1).
The first few polynomials are
P_1(x) = x-2
P_2(x) = x^2-6*x+5
P_3(x) = x^3-12*x^2+32*x-12.
It appears that a(n+1) = P_n(1) (checked as far as a(19)).
Compare with A144010.

			E.g.f.: A(x) = 1 + x - x^2/2! + 9*x^4/4! - 48*x^5/5! + 15*x^6/6! + 2448*x^7/7! +...
where
1/(1 + x*A(x)) = 1 - x + 9*x^3/3! - 48*x^4/4! + 15*x^5/5! + 2448*x^6/6! +...
Also, A(G(x)) = 1 + x where
G(x) = x + x^2/2! + 3*x^3/3! + 6*x^4/4! + 18*x^5/5! + 48*x^6/6! + 156*x^7/7! + 492*x^8/8! +...+ A000932(n-1)*x^n/n! +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..190

Cf. A144010, A000932.

m = 20; A[_] = 1;
Do[A[x_] = 1 + Integrate[1/(1+x*A[x])+O[x]^m, x]+O[x]^m // Normal, {m}];
CoefficientList[A[x], x] * Range[0, m-1]! (* Jean-François Alcover, Nov 02 2019 *)

{a(n)=local(A=1+x); for(i=0, n, A=1+intformal(1/(1+x*A+x*O(x^n)) ));n!*polcoeff(A, n)}

E.g.f. satisfies: A(x) = 1 + Integral 1/(1 + x*A(x)) dx.

E.g.f. satisfies: A(G(x)) = 1 + x where G(x) is the e.g.f. of A000932 (offset 1). [Paul D. Hanna, Aug 23 2011]

A238302 E.g.f. satisfies: A'(x) = 1/(1 - x*A(x))^3 with A(0)=1.

Original entry on oeis.org

1, 1, 3, 18, 159, 1872, 27585, 488736, 10122273, 240029568, 6413759739, 190698235200, 6244960476447, 223354435468032, 8662843993599081, 362162307560207616, 16235331377066437185, 776899798554908971008, 39526189786699908619635, 2130538920122365706916864

Offset: 0

Vaclav Kotesovec, Feb 24 2014

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..277

Cf. A144010, A144011.

{a(n)=local(A=1+x); for(i=0, n, A=1+intformal(1/(1-x*A+x*O(x^n))^3 )); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

a(n) ~ c * n! * d^n / n^(5/4), where c = 0.502313... and d = 3.03113979...

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A144011 E.g.f. satisfies: A'(x) = 1/(1 - x*A(x))^2 with A(0)=1.

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A173895 E.g.f. satisfies: A'(x) = 1/(1 + x*A(x)) with A(0) = 1.

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A238302 E.g.f. satisfies: A'(x) = 1/(1 - x*A(x))^3 with A(0)=1.

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