A213643 E.g.f. satisfies: A(x) = x + A(x)^2*exp(A(x)).
1, 2, 18, 252, 4940, 124350, 3823722, 138915560, 5822192952, 276522143130, 14677209803630, 860990013672492, 55315008281020644, 3862656545279925302, 291301089508829138130, 23595204076694940812880, 2042970533426395737658352, 188298566037963463789282482
Offset: 1
Keywords
Examples
E.g.f.: A(x) = x + 2*x^2/2! + 18*x^3/3! + 252*x^4/4! + 4940*x^5/5! +... where A(x - x^2*exp(x)) = x and A(x) = x + A(x)^2*exp(A(x)). Related expansions: A(x)^2 = 2*x^2/2! + 12*x^3/3! + 168*x^4/4! + 3240*x^5/5! + 80880*x^6/6! +... A(x) = x*Catalan(x*G(x)) where G(x) = exp(A(x)): exp(A(x)) = 1 + x + 3*x^2/2! + 25*x^3/3! + 349*x^4/4! + 6821*x^5/5! + 171421*x^6/6! +..., which is the e.g.f. of A161629. A(x) = x + exp(x)*x^2 + d/dx exp(2*x)*x^4/2! + d^2/dx^2 exp(3*x)*x^6/3! + d^3/dx^3 exp(4*x)*x^8/4! +... log(A(x)/x) = exp(x)*x + d/dx exp(2*x)*x^3/2! + d^2/dx^2 exp(3*x)*x^5/3! + d^3/dx^3 exp(4*x)*x^7/4! +... Ordinary Generating Function: O.g.f.: x + 2*x^2 + 18*x^3 + 252*x^4 + 4940*x^5 + 124350*x^6 +... O.g.f.: x + 2*x^2/(1-x)^3 + 6*2!*x^3/(1-2*x)^5 + 20*3!*x^4/(1-3*x)^7 + 70*4!*x^5/(1-4*x)^9 + 252*5!*x^6/(1-5*x)^11 +...+ (2*n)!/n!*x^(n+1)/(1-n*x)^(2*n+1) +...
Links
- Vincenzo Librandi, Table of n, a(n) for n = 1..200
Programs
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Maple
a:= n-> n!*coeff(series(RootOf(A=x+A^2*exp(A), A), x, n+1), x, n): seq(a(n), n=0..30); # Alois P. Heinz, Jul 18 2013
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Mathematica
Flatten[{1,Table[Sum[k^(n-k-1)/(n-k-1)!*(n+k-1)!/k!,{k,0,n-1}],{n,2,20}]}] (* Vaclav Kotesovec, Jul 13 2013 *) -
PARI
{a(n)=sum(k=0,n-1, k^(n-k-1)/(n-k-1)! * (n+k-1)!/k! )} -
PARI
{a(n)=n!*polcoeff(serreverse(x-x^2*exp(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(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(m*x+x*O(x^n))*x^(2*m-1)/m!)+x*O(x^n))); n!*polcoeff(A, n)} -
PARI
/* O.g.f.: */ {a(n)=polcoeff(sum(m=0,n,(2*m)!/m!*x^(m+1)/(1-m*x+x*O(x^n))^(2*m+1)),n)}
Formula
E.g.f.: A(x) = log(G(x)) where G(x) = exp(x*Catalan(x*G(x))) is the e.g.f. of A161629, and Catalan(x) = (1-sqrt(1-4*x))/(2*x).
E.g.f.: Series_Reversion(x - x^2*exp(x)).
E.g.f.: x + Sum_{n>=1} d^(n-1)/dx^(n-1) exp(n*x)*x^(2*n) / n!.
E.g.f.: x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) exp(n*x)*x^(2*n-1) / n! ).
O.g.f.: Sum_{n>=0} (2*n)!/n! * x^(n+1) / (1 - n*x)^(2*n+1).
a(n) = Sum_{k=0..n-1} k^(n-k-1)/(n-k-1)! * (n+k-1)!/k!.
a(n) = n*A213644(n-1).
Limit n->infinity (a(n)/n!)^(1/n) = r*(1+r)/(1-r) = 5.5854662015218413..., where r = 0.7603592340333989... is the root of the equation (1-r^2)/r^2 = exp((r-1)/r). - Vaclav Kotesovec, Jul 13 2013
a(n) ~ (1-r) * n^(n-1) * (r*(1+r)/(1-r))^n / (sqrt(r*(1+2*r-r^2))*exp(n)). - Vaclav Kotesovec, Dec 28 2013