A214688 - OEIS

%I #22 Mar 09 2024 01:28:18

%S 1,2,24,408,9760,299520,11223744,496802432,25365482496,1467480983040,

%T 94873742909440,6778628603670528,530412734126346240,

%U 45110083291805622272,4143219058165730672640,408715543077297795072000,43097868598208296895512576,4837629293480336802779234304

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

%H Vincenzo Librandi, <a href="/A214688/b214688.txt">Table of n, a(n) for n = 1..200</a>

%F a(n) = Sum_{k=0..n-1} ( (n+k-1)! * (2*k)^(n-k-1) )/(k! * (n-k-1)!).

%F O.g.f.: Sum_{n>=0} (2*n)!/n! * x^(n+1) / (1 - 2*n*x)^(2*n+1).

%F E.g.f. satisfies:

%F (1) A(x) = x + A(x)^2*exp(2*A(x)).

%F (2) A(x) = x*Catalan( x*exp(2*A(x)) ) where Catalan(x) = (1-sqrt(1-4*x))/(2*x).

%F (3) A(x) = x*Sum_{n>=0} binomial(2*n+1,n)/(2*n+1) * x^n * exp(2*n*A(x)).

%F (4) A(x) = x*exp( Sum_{n>=1} binomial(2*n-1,n) * x^n * exp(2*n*A(x)) / n ).

%F (5) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n) * exp(2*n*x) / n!.

%F (6) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n-1) * exp(2*n*x) / n! ).

%F (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).

%F 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

%F 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

%e E.g.f.: A(x) = x + 2*x^2/2! + 24*x^3/3! + 408*x^4/4! + 9760*x^5/5! + ...

%e where A(x - x^2*exp(2*x)) = x and A(x) = x + A(x)^2*exp(2*A(x)).

%e The e.g.f. satisfies:

%e (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)) + ...

%e (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 + ...

%e (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! + ...

%e (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! + ...

%e Related expansions:

%e A(x) = x*Catalan(x*G(x)) where G(x) = exp(2*A(x)):

%e 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! + ...

%e 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.

%e A(x)^2 = 2*x^2/2! + 12*x^3/3! + 216*x^4/4! + 5040*x^5/5! + 153120*x^6/6! + ...

%e Ordinary Generating Function:

%e O.g.f.: x + 2*x^2 + 24*x^3 + 408*x^4 + 9760*x^5 + 299520*x^6 + ...

%e 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) + ...

%t 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 *)

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

%o (PARI) {a(n)=n!*polcoeff(serreverse(x-x^2*exp(2*x+x*O(x^n))), n)}

%o for(n=1, 25, print1(a(n), ", "))

%o (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}

%o {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)}

%o (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}

%o {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)}

%o (PARI) /* From o.g.f.: */

%o {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)}

%o (Magma)

%o 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]]) >;

%o [A214688(n): n in [1..30]]; // _G. C. Greubel_, Mar 07 2024

%o (SageMath)

%o 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))

%o [A214688(n) for n in range(1,30)] # _G. C. Greubel_, Mar 07 2024

%Y Cf. A213643, A214689, A215003.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Aug 01 2012

%E Name changed and entry revised by _Paul D. Hanna_, Jul 13 2013