A138293 E.g.f. satisfies: A(x) = x + exp(x*A(x)).
1, 2, 5, 28, 245, 2816, 40537, 702976, 14270153, 332102656, 8719631981, 255020847104, 8222803663549, 289815184113664, 11085650268060929, 457386463819595776, 20248713707077863953, 957435459515190345728
Offset: 0
Keywords
Examples
E.g.f: A(x) = 1 + 2*x + 5/2*x^2 + 14/3*x^3 + 245/24*x^4 + 352/15*x^5 +... Log(A(x)-x) = x + 2*x^2 + 5/2*x^3 + 14/3*x^4 + 245/24*x^5 + 352/15*x^6 +...
Programs
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Maple
A138293 := proc(n) add( (n-2*k+1)^(n-k)/k!/(n-2*k+1)!,k=0..(n+1)/2) ; %*n! ; end proc: seq(A138293(n),n=0..30) ; # R. J. Mathar, May 03 2023
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Mathematica
nmax = 20; CoefficientList[Series[x - ProductLog[-x*E^(x^2)]/x, {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Nov 21 2017 *) -
PARI
{a(n)=local(A=1);for(i=0,n,A=x+exp(x*A+x*O(x^n)));n!*polcoeff(A,n)}
Formula
a(n) = n!*Sum_{k=0..floor((n+1)/2)} (n-2*k+1)^(n-k)/(k!*(n-2*k+1)!). - Vladeta Jovovic, Mar 15 2008
a(n) ~ sqrt(1 + LambertW(2*exp(-2))) * (2/LambertW(2*exp(-2)))^((n+1)/2) * n^(n-1) / exp(n). - Vaclav Kotesovec, Nov 21 2017