|
| |
|
|
A214688
|
|
E.g.f. equals the series reversion of x - x^2*exp(2*x).
|
|
2
|
|
|
|
1, 2, 24, 408, 9760, 299520, 11223744, 496802432, 25365482496, 1467480983040, 94873742909440, 6778628603670528, 530412734126346240, 45110083291805622272, 4143219058165730672640, 408715543077297795072000, 43097868598208296895512576, 4837629293480336802779234304
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,2
|
|
|
LINKS
|
|
|
|
FORMULA
|
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
|
|
|
EXAMPLE
|
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) + ...
|
|
|
MATHEMATICA
|
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 *)
|
|
|
PROG
|
(PARI) {a(n)=sum(k=0, n-1, (2*k)^(n-k-1)/(n-k-1)! * (n+k-1)!/k! )}
(PARI) {a(n)=n!*polcoeff(serreverse(x-x^2*exp(2*x+x*O(x^n))), n)}
for(n=1, 25, print1(a(n), ", "))
(PARI) {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)}
(PARI) {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)}
(PARI) /* 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)}
(Magma)
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]]) >;
(SageMath)
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))
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
|
|
|
STATUS
|
approved
|
| |
|
|
