A238302 -id:A238302 - cp's OEIS frontend

A144010 E.g.f. satisfies: A'(x) = 1/(1 - x*A(x)) with A(0)=1.

1, 1, 1, 4, 21, 160, 1525, 17760, 243145, 3833600, 68373225, 1361264000, 29925477725, 719991897600, 18817847565725, 530921477363200, 16082605690148625, 520603130117939200, 17934634668874889425

Offset: 0

Paul D. Hanna, Sep 10 2008

nonn

From Peter Bala, Nov 26 2010: (Start)
Define a polynomial sequence P_n(x) recursively by
... P_0(x) = 1,
... P_n(x) = (x-1)*P_(n-1)(x-1) + n*P_(n-1)(x+1) for n >= 1.
The first few polynomials are
P_1(x) = x;
P_2(x) = x^2 + 3;
P_3(x) = x^3 + 12*x + 8.
It appears that a(n+1) = P_n(1) (checked as far as a(19)).
Compare with A173895. (End)

Vaclav Kotesovec, Table of n, a(n) for n = 0..400

Cf. A144011, A238302.

FindRoot[Sqrt[Pi/2]*s*E^(-s^2/2)*(Erfi[1/Sqrt[2]]-Erfi[s/Sqrt[2]]) == -1,{s,1},WorkingPrecision->50] (* program for numerical value of the constant s, Vaclav Kotesovec, Feb 23 2014 *)
a[0] = 1; a[1] = 1; a[n_] := a[n] = Sum[k Binomial[n-1, k] a[k] a[n-k-1], {k, 1, n-1}]; Table[a[n], {n, 0, 20}] (* Vladimir Reshetnikov, May 17 2016 *)

{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)}
for(n=0,30,print1(a(n),", "))

/* from Peter Bala's Formula */
{a(n)=local(P=1);if(n>=0&n<2,1,for(k=1,n-1,P=(x-1)*subst(P,x,x-1) + k*subst(P,x,x+1)));subst(P,x,1)}
for(n=0,30,print1(a(n),", ")) \\ Paul D. Hanna, Jun 15 2013

E.g.f. satisfies: A(x) = 1 + Integral 1/(1 - x*A(x)) dx.
a(n) ~ n^(n-1) * s^n / exp(n), where s = 2.0832144900084392272885741721727173082215... is the root of the equation sqrt(Pi/2)*s*exp(-s^2/2)*(erfi(1/sqrt(2)) - erfi(s/sqrt(2))) = -1. - Vaclav Kotesovec, Feb 23 2014
a(0) = 1, a(1) = 1, a(n) = Sum_{0 < k < n} k * binomial(n-1, k) * a(k) * a(n-k-1). - Vladimir Reshetnikov, May 17 2016

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

Original entry on oeis.org

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

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A144010 E.g.f. satisfies: A'(x) = 1/(1 - x*A(x)) with A(0)=1.

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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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