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A143155 E.g.f.: A(x) = -log(1 - x - A(x)^2). 1
1, 3, 26, 376, 7614, 198248, 6309092, 237291388, 10297903920, 506495785632, 27842563031304, 1691646018671376, 112569103111005072, 8142200129607522288, 636046143210331062048, 53366672768969064921024 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Table of n, a(n) for n=1..16.

FORMULA

a(n) = Sum_{k=0..n-1} (n+k-1)!*Sum_{j=0..k} ((-1)^(n+j-1)/(k-j)!)*Sum_{L=0..min(j, (n+j-1)/2)} Stirling2(n-2*L+j-1, j-l)/(L!*(n-2*L+j-1)!), n > 0. - Vladimir Kruchinin, Feb 03 2012

a(n) ~ n^(n-1) / (sqrt(2*(1+c)) * exp(n) * (1-2*c-c^2)^(n-1/2)), where c = LambertW(1/2). - Vaclav Kotesovec, Dec 28 2013

EXAMPLE

A(x) = x + 3*x^2/2! + 26*x^3/3! + 376*x^4/4! + 7614*x^5/5! +...

x + A(x)^2 = 1 - exp(-A(x)) = G(x) = g.f. of A143154:

G(x) = x + 2*x^2/2! + 18*x^3/3! + 262*x^4/4! + 5320*x^5/5! +...

A(x)^2 = 2*x^2/2! + 18*x^3/3! + 262*x^4/4! + 5320*x^5/5! +...

MATHEMATICA

Table[Sum[(n+k-1)!*Sum[(-1)^(n+j-1)/(k-j)!*Sum[(StirlingS2[n-2*l+j-1, j-l])/(l!*(n-2*l+j-1)!), {l, 0, Min[j, (n+j-1)/2]}], {j, 0, k}], {k, 0, n-1}], {n, 1, 20}] (* Vaclav Kotesovec after Vladimir Kruchinin, Dec 28 2013 *)

PROG

(PARI) {a(n)=local(A=x+O(x^n)); for(i=0, n, A=-log(1-x-A^2)); n!*polcoeff(A, n)}

(PARI) {a(n)=n!*polcoeff(-log(1-serreverse(x-log(1-x+x*O(x^n))^2)), n)}

(Maxima) a(n):=(sum((n+k-1)!*sum((-1)^(n+j-1)/(k-j)!*sum((stirling2(n-2*l+j-1, j-l))/(l!*(n-2*l+j-1)!), l, 0, min(j, (n+j-1)/2)), j, 0, k), k, 0, n-1)); /* Vladimir Kruchinin, Feb 03 2012 */

CROSSREFS

Cf. A143139, A143154, A202356.

Sequence in context: A206404 A262301 A317654 * A300283 A326431 A206403

Adjacent sequences:  A143152 A143153 A143154 * A143156 A143157 A143158

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jul 27 2008

STATUS

approved

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Last modified January 19 18:52 EST 2022. Contains 350466 sequences. (Running on oeis4.)