A214689 -id:A214689 - cp's OEIS frontend

A161629 E.g.f. satisfies: A(x) = exp( xCatalan(xA(x)) ), where Catalan(x) = (1-sqrt(1-4x))/(2x) is the g.f. of A000108.

1, 1, 3, 25, 349, 6821, 171421, 5265625, 191160201, 8007548617, 380157603481, 20171371753061, 1182973489103869, 75984447924612397, 5305029326492409333, 400014338565211619761, 32396515980658185762961

Offset: 0

Paul D. Hanna, Jun 17 2009

nonn

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 25*x^3/3! + 349*x^4/4! + 6821*x^5/5! +...
log(A(x))/x = 1 + x*A(x) + 2*x^2*A(x)^2 + 5*x^3*A(x)^3 + 14*x^4*A(x)^4 +...+ A000108(n)*x^n*A(x)^n +...
log(A(x))/x = 1 + x + 6*x^2/2! + 63*x^3/3! + 988*x^4/4! + 20725*x^5/5! +...+ A213644(n)*x^n/n! +...
log(A(x)) = x + 2*x^2/2! + 18*x^3/3! + 252*x^4/4! + 4940*x^5/5! +...+ A213643(n)*x^n/n! +...
Ordinary Generating Function:
O.g.f.: 1 + x + 3*x^2 + 25*x^3 + 349*x^4 + 6821*x^5 + 171421*x^6 +...
O.g.f.: 1 + x/(1-x) + 2*x^2/(1-2*x)^3 + 6*2!*x^3/(1-3*x)^5 + 20*3!*x^4/(1-4*x)^7 + 70*4!*x^5/(1-5*x)^9 + 252*5!*x^6/(1-6*x)^11 +...+ (2*n-2)!/(n-1)!*x^n/(1-n*x)^(2*n-1) +...

Cf. A213643 (log), A214689, A000108.

Flatten[{1,Table[Sum[n!/k!*(n-k+1)^(k-1)*Binomial[2*n-k, n-k]*k/(2*n-k),{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, Feb 26 2014 *)

{a(n,m=1)=if(n==0,1,sum(k=0,n,n!/k!*m*(m+n-k)^(k-1)*binomial(2*n-k,n-k)*k/(2*n-k)))}

{a(n,m=1)=local(A=1+x+x*O(x^n));for(i=1,n,A=exp((1-sqrt(1-4*x*A))/(2*A)));n!*polcoeff(A^m,n)}

/* O.g.f.: */
{a(n)=polcoeff(1+sum(m=1, n, (2*m-2)!/(m-1)!*x^m/(1-m*x+x*O(x^n))^(2*m-1)), n)}

E.g.f.: A(x) = exp(F(x)) where F(x) = x + F(x)^2*exp(F(x)) is the e.g.f. of A213643.
E.g.f.: A(x) = Sum_{n>=0} a(n)*x^n/n!, where
a(n) = Sum_{k=0..n} n! * (n-k+1)^(k-1)/k! * C(2*n-k,n-k)*k/(2*n-k).
Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n!, then
a(n,m) = Sum_{k=0..n} n! * m*(n-k+m)^(k-1)/k! * C(2*n-k,n-k)*k/(2*n-k).
...
O.g.f.: A(x) = 1 + Sum_{n>=1} (2*n-2)!/(n-1)! * x^n/(1 - n*x)^(2*n-1).
a(n) ~ n^(n-1) * sqrt((r*s^3*(1-6*r*s+8*r^2*s^2)) / (1 - (-2+8*r+r^2)*s + 4*r*(-4+4*r+r^2)*s^2 + 4*r^2*(8+r)*s^3)) / (exp(n) * r^n), where s = 1.370489293947401403417767032... is the root of the equation log(s)*(1-s*log(s)) + 2*(1+s) = (1+2*s) * sqrt((1+s)/s), and r = log(s)*(1-s*log(s)) = 0.179036084709909351719214... - Vaclav Kotesovec, Feb 26 2014

A214688 E.g.f. equals the series reversion of x - x^2exp(2x).

Original entry on oeis.org

1, 2, 24, 408, 9760, 299520, 11223744, 496802432, 25365482496, 1467480983040, 94873742909440, 6778628603670528, 530412734126346240, 45110083291805622272, 4143219058165730672640, 408715543077297795072000, 43097868598208296895512576, 4837629293480336802779234304

Offset: 1

Paul D. Hanna, Aug 01 2012

nonn

			E.g.f.: A(x) = x + 2*x^2/2! + 24*x^3/3! + 408*x^4/4! + 9760*x^5/5! + ...
where A(x - x^2*exp(2*x)) = x and A(x) = x + A(x)^2*exp(2*A(x)).
The e.g.f. satisfies:
(3) A(x) = x + x^2*exp(2*A(x)) + 2*x^3*exp(4*A(x)) + 5*x^4*exp(6*A(x)) + 14*x^5*exp(8*A(x)) + 42*x^6*exp(10*A(x)) + ...
(4) log(A(x)/x) = x*exp(2*A(x)) + 3*x^2*exp(4*A(x))/2 + 10*x^3*exp(6*A(x))/3 + 35*x^4*exp(8*A(x))/4 + 126*x^5*exp(10*A(x))/5 + ...
(5) A(x) = x + x^2*exp(2*x) + d/dx x^4*exp(4*x)/2! + d^2/dx^2 x^6*exp(6*x)/3! + d^3/dx^3 x^8*exp(8*x)/4! + ...
(6) log(A(x)/x) = x*exp(2*x) + d/dx x^3*exp(4*x)/2! + d^2/dx^2 x^5*exp(6*x)/3! + d^3/dx^3 x^7*exp(8*x)/4! + ...
Related expansions:
A(x) = x*Catalan(x*G(x)) where G(x) = exp(2*A(x)):
exp(A(x)) = 1 + x + 3*x^2/2! + 31*x^3/3! + 529*x^4/4! + 12601*x^5/5! + 385891*x^6/6! + ...
exp(2*A(x)) = 1 + 2*x + 8*x^2/2! + 80*x^3/3! + 1360*x^4/4! + 32352*x^5/5! + 989824*x^6/6! + ..., which is the e.g.f. of A214689.
A(x)^2 = 2*x^2/2! + 12*x^3/3! + 216*x^4/4! + 5040*x^5/5! + 153120*x^6/6! + ...
Ordinary Generating Function:
O.g.f.: x + 2*x^2 + 24*x^3 + 408*x^4 + 9760*x^5 + 299520*x^6 + ...
O.g.f.: x + 2*x^2/(1-2*x)^3 + 6*2!*x^3/(1-4*x)^5 + 20*3!*x^4/(1-6*x)^7 + 70*4!*x^5/(1-8*x)^9 + 252*5!*x^6/(1-10*x)^11 + ... + (2*n)!/n!*x^(n+1)/(1-2*n*x)^(2*n+1) + ...

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A213643, A214689, A215003.

A214688:= func< n | n eq 1 select 1 else (&+[Binomial(n-1,k)*Binomial(n+k-1,k)*Factorial(k)*(2*k)^(n-k-1): k in [1..n-1]]) >;
[A214688(n): n in [1..30]]; // G. C. Greubel, Mar 07 2024

Flatten[{1,Table[Sum[(2*k)^(n-k-1)/(n-k-1)!*(n+k-1)!/k!,{k,1,n-1}],{n,2,20}]}] (* Vaclav Kotesovec, Jul 13 2013 *)

{a(n)=sum(k=0, n-1, (2*k)^(n-k-1)/(n-k-1)! * (n+k-1)!/k! )}

{a(n)=n!*polcoeff(serreverse(x-x^2*exp(2*x+x*O(x^n))), n)}
for(n=1, 25, print1(a(n), ", "))

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, exp(2*m*x+x*O(x^n))*x^(2*m)/m!)); n!*polcoeff(A, n)}

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, exp(2*m*x+x*O(x^n))*x^(2*m-1)/m!)+x*O(x^n))); n!*polcoeff(A, n)}

/* From o.g.f.: */
{a(n)=polcoeff(sum(m=0, n, (2*m)!/m!*x^(m+1)/(1-2*m*x+x*O(x^n))^(2*m+1)), n)}

def A214688(n): return int(n==1)+sum(binomial(n-1,k)*binomial(n+k-1,k)*factorial(k)*(2*k)^(n-k-1) for k in range(1,n))
[A214688(n) for n in range(1,30)] # G. C. Greubel, Mar 07 2024

a(n) = Sum_{k=0..n-1} ( (n+k-1)! * (2*k)^(n-k-1) )/(k! * (n-k-1)!).
O.g.f.: Sum_{n>=0} (2*n)!/n! * x^(n+1) / (1 - 2*n*x)^(2*n+1).
E.g.f. satisfies:
(1) A(x) = x + A(x)^2*exp(2*A(x)).
(2) A(x) = x*Catalan( x*exp(2*A(x)) ) where Catalan(x) = (1-sqrt(1-4*x))/(2*x).
(3) A(x) = x*Sum_{n>=0} binomial(2*n+1,n)/(2*n+1) * x^n * exp(2*n*A(x)).
(4) A(x) = x*exp( Sum_{n>=1} binomial(2*n-1,n) * x^n * exp(2*n*A(x)) / n ).
(5) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n) * exp(2*n*x) / n!.
(6) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n-1) * exp(2*n*x) / n! ).
(7) A(x) = log(G(x))/2 where G(x) = exp(2*x*Catalan(x*G(x))) is the e.g.f. of A214689, and Catalan(x) = (1-sqrt(1-4*x))/(2*x).
Limit_{n->oo} (a(n)/n!)^(1/n) = (2*r*(1+r))/(1-r) = 6.801725926701655517492664481..., where r = 0.670589533381613711... is the root of the equation (1-r^2)/(2*r^2) = exp((r-1)/r). - Vaclav Kotesovec, Jul 13 2013
a(n) ~ (1-r) * n^(n-1) * (2*r*(1+r)/(1-r))^n / (2*sqrt(r*(1+2*r-r^2)) * exp(n)). - Vaclav Kotesovec, Dec 28 2013

Name changed and entry revised by Paul D. Hanna, Jul 13 2013

A382029 E.g.f. A(x) satisfies A(x) = exp(xC(xA(x)^2)), where C(x) = 1 + x*C(x)^2 is the g.f. of A000108.

Original entry on oeis.org

1, 1, 3, 31, 529, 12601, 385891, 14440567, 638576065, 32580927505, 1883889232291, 121742057314351, 8695278706372369, 680187946863332233, 57833833258995140803, 5310742450917819399751, 523793286672328763358721, 55223769332070053104438945, 6197871354601209094032190147

Offset: 0

Seiichi Manyama, Mar 12 2025

nonn

Cf. A212722, A382030, A382031.
Cf. A161629, A382042.
Cf. A000108, A214688, A214689, A379690.

a(n) = if(n==0, 1, n!*sum(k=0, n-1, (2*k+1)^(n-k-1)*binomial(n+k, k)/((n+k)*(n-k-1)!)));

Let F(x) be the e.g.f. of A379690. F(x) = log(A(x))/x = C(x*A(x)^2).
E.g.f.: A(x) = exp( Series_Reversion( x*(1 - x*exp(2*x)) ) ).
a(n) = n! * Sum_{k=0..n-1} (2*k+1)^(n-k-1) * binomial(n+k,k)/((n+k) * (n-k-1)!) for n > 0.

cp's OEIS Frontend

A161629 E.g.f. satisfies: A(x) = exp( xCatalan(xA(x)) ), where Catalan(x) = (1-sqrt(1-4x))/(2x) is the g.f. of A000108.

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A214688 E.g.f. equals the series reversion of x - x^2exp(2x).

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A382029 E.g.f. A(x) satisfies A(x) = exp(xC(xA(x)^2)), where C(x) = 1 + x*C(x)^2 is the g.f. of A000108.

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cp's OEIS Frontend

A161629 E.g.f. satisfies: A(x) = exp( x*Catalan(x*A(x)) ), where Catalan(x) = (1-sqrt(1-4*x))/(2*x) is the g.f. of A000108.

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A214688 E.g.f. equals the series reversion of x - x^2*exp(2*x).

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A382029 E.g.f. A(x) satisfies A(x) = exp(x*C(x*A(x)^2)), where C(x) = 1 + x*C(x)^2 is the g.f. of A000108.

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A161629 E.g.f. satisfies: A(x) = exp( xCatalan(xA(x)) ), where Catalan(x) = (1-sqrt(1-4x))/(2x) is the g.f. of A000108.

A214688 E.g.f. equals the series reversion of x - x^2exp(2x).

A382029 E.g.f. A(x) satisfies A(x) = exp(xC(xA(x)^2)), where C(x) = 1 + x*C(x)^2 is the g.f. of A000108.