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A143740 E.g.f. satisfies A(x) = exp(x + x^2/2 * A(x)). 6
1, 1, 2, 7, 34, 216, 1696, 15898, 173468, 2161036, 30282076, 471599316, 8082816160, 151218316120, 3066890630168, 67031194526416, 1570793031033616, 39290173530686544, 1044871388684004304, 29440090627527552976 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Eric Weisstein's World of Mathematics, Lambert W-Function.
FORMULA
E.g.f.: A(x) = -2*LambertW( -x^2*exp(x)/2 )/x^2.
E.g.f.: A(x) = Sum_{n>=0} (n+1)^(n-1)*(x^2/2)^n*exp((n+1)*x)/n!.
a(n) ~ sqrt(1+LambertW(1/sqrt(2*exp(1)))) * n^(n-1) /(2^(n+1/2) * exp(n) * (LambertW(1/sqrt(2*exp(1))))^(n+2)). - Vaclav Kotesovec, Jul 09 2013
Recurrence: a(0)=1, a(1)=1, for n > 1, a(n) = a(n-1) + Sum_{k=0..n-2} (n-1)*(n-k)*binomial(n-2,k)*a(k)*a(n-2-k)/2. - Tani Akinari, Nov 01 2017
From Seiichi Manyama, Apr 20 2023: (Start)
E.g.f.: exp(x - LambertW(-x^2/2 * exp(x))).
a(n) = n! * Sum_{k=0..floor(n/2)} (1/2)^k * (k+1)^(n-k-1) / (k! * (n-2*k)!). (End)
EXAMPLE
E.g.f.: A(x) = 1 + x + 2*x^2/2! + 7*x^3/3! + 34*x^4/4! + 216*x^5/5! + ...
MATHEMATICA
CoefficientList[Series[-2*LambertW[-x^2*E^x/2]/x^2, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jul 09 2013 *)
PROG
(PARI) {a(n)=local(A=1+x*O(x^n)); for(i=0, n, A=exp(x+x^2*A/2)); (n+0)!*polcoeff(A, n)}
(PARI) {a(n)=local(A=sum(m=0, n, (m+1)^(m-1)*(x^2/2)^m*exp((m+1)*x+x*O(x^n))/m!)); n!*polcoeff(A, n)}
(PARI) {a(n)=local(Ex=exp(x+x*O(x^n)), W=Ex); for(k=0, n, W=exp(x*W)); n!*polcoeff(subst(W, x, x^2*Ex/2)*Ex, n)}
(Maxima) a[n]:=(if n<2 then 1 else a[n-1]+sum((n-1)*(n-k)*binomial(n-2, k)*a[k]*a[n-2-k], k, 0, n-2)/2);
makelist(a[n], n, 0, 100); /* Tani Akinari, Nov 01 2017 */
CROSSREFS
Column k=1 of A362377.
Cf. A125500.
Sequence in context: A171792 A185324 A135882 * A049463 A294466 A029894
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Aug 30 2008
STATUS
approved

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Last modified March 28 07:33 EDT 2024. Contains 371235 sequences. (Running on oeis4.)