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A125500
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Expansion of -LambertW(-x^2*exp(x))/x^2.
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4
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1, 1, 3, 13, 85, 701, 7261, 89125, 1277865, 20883385, 384194521, 7852225481, 176651705869, 4337650936789, 115468033349397, 3312409332578221, 101881034223806161, 3344745711740899697, 116747433680684736817
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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FORMULA
| E.g.f.: A(x) = exp(x + x^2*A(x)). [From Paul D. Hanna (pauldhanna(AT)juno.com), Aug 30 2008]
Contribution from Paul D. Hanna (pauldhanna(AT)juno.com), Jun 17 2009: (Start)
a(n) = Sum_{k=0..n} n! * (n-k+1)^(k-1)/k! * C(k,n-k).
Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n!, then
a(n,m) = Sum_{k=0..n} n! * m*(n-k+m)^(k-1)/k! * C(k,n-k). (End)
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EXAMPLE
| E.g.f.: A(x) = 1 + x + 3*x^2/2! + 13*x^3/3! + 85*x^4/4! + 701*x^5/5! +... [From Paul D. Hanna (pauldhanna(AT)juno.com), Aug 30 2008]
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PROG
| (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)*Ex, n)} - Paul D. Hanna (pauldhanna(AT)juno.com), Jan 02 2007
(PARI) {a(n)=local(A=1+x*O(x^n)); for(i=0, n, A=exp(x+x^2*A)); n!*polcoeff(A, n)} [From Paul D. Hanna (pauldhanna(AT)juno.com), Aug 30 2008]
Contribution from Paul D. Hanna (pauldhanna(AT)juno.com), Jun 17 2009: (Start)
(PARI) {a(n, m=1)=if(n==0, 1, sum(k=0, n, n!/k!*m*(n-k+m)^(k-1)*binomial(k, n-k)))}
(PARI) {a(n, m=1)=local(A=1+x+x*O(x^n)); for(i=1, n, A=exp(x*(1+x*A))); n!*polcoeff(A^m, n)} (End)
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CROSSREFS
| Sequence in context: A130406 A152789 A192943 * A121679 A023037 A157451
Adjacent sequences: A125497 A125498 A125499 * A125501 A125502 A125503
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KEYWORD
| easy,nonn
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AUTHOR
| Vladeta Jovovic (vladeta(AT)eunet.rs), Dec 27 2006
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EXTENSIONS
| More terms from Paul D. Hanna (pauldhanna(AT)juno.com), Jan 02 2007
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