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A193290 E.g.f. satisfies: A(x) = 1/(1 - x*A(x))^(1 + 1/A(x)). 5
1, 2, 10, 96, 1388, 26960, 659352, 19471984, 674425600, 26814697056, 1203912012000, 60251644584384, 3326134996826688, 200792710948417536, 13159474030202943744, 930524202271542658560, 70616227020854238216192, 5724780985202503068533760 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

More generally, we have the identity:

Sum_{n>=0} (x^n/n!)*Product_{k=1..n} (1+k*y) = 1/(1 - x*y)^(1 + 1/y); here y=A(x).

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..100

FORMULA

E.g.f. satisfies: A(x) = Sum_{n>=0} x^n/n! * Product_{k=1..n} (1 + k*A(x)).

a(n) ~ s*sqrt(r*(1+s)*(r*s-1)/(1-4*r*s+r^2*s*(2*s-1))) * n^(n-1) / (exp(n) * r^n), where s = 2.4590533113276368838... is the root of the equation (1+s)*(1+2*s) = s^(s/(1+s))*(1+s)^2 - s*log(s) and r = (1 - s^(-s/(1+s)))/s = 0.1921573821382919835... - Vaclav Kotesovec, Jan 11 2014

EXAMPLE

E.g.f.: A(x) = 1 + 2*x + 10*x^2/2! + 96*x^3/3! + 1388*x^4/4! + 26960*x^5/5! +...

where e.g.f. A = A(x) satisfies:

A = 1 + x*(1+A) + x^2*(1+A)*(1+2*A)/2! + x^3*(1+A)*(1+2*A)*(1+3*A)/3! + x^4*(1+A)*(1+2*A)*(1+3*A)*(1+4*A)/4! + x^5*(1+A)*(1+2*A)*(1+3*A)*(1+4*A)*(1+5*A)/5! +...

PROG

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1/(1-x*A +x*O(x^n))^(1+1/A)); n!*polcoeff(A, n)}

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, x^m/m!*prod(k=1, m, 1+k*A+x*O(x^n)))); n!*polcoeff(A, n)}

CROSSREFS

Cf. A094638, A193289, A193281, A193287, A193288.

Sequence in context: A103436 A182173 A160940 * A193435 A132572 A069247

Adjacent sequences:  A193287 A193288 A193289 * A193291 A193292 A193293

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jul 21 2011

STATUS

approved

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Last modified February 28 01:28 EST 2020. Contains 332319 sequences. (Running on oeis4.)