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A162655
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E.g.f. satisfies: A(x) = (1 + x*A(x))^A(x).
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3
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1, 1, 4, 33, 416, 7100, 153234, 4004000, 122919208, 4336955424, 172946624880, 7692618593352, 377615317473624, 20278301717340888, 1182581903027279832, 74428445506232769240, 5028336618916834615104, 362962785521720282899200
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Contribution from Paul D. Hanna (pauldhanna(AT)juno.com), Jul 19 2009: (Start)
More generally, if G(x) = (1 + x*G(x)^p)^(G(x)^q), then
[x^n/n! ] G(x)^m = Sum_{k=0..n} m*(pn+qk+m)^(k-1) * Stirling1(n,k), and
[x^n/n! ] log(G(x)) = Sum_{k=1..n} (pn+qk)^(k-1) * Stirling1(n,k). (End)
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FORMULA
| (1) a(n) = Sum_{k=0..n} (n+k+1)^(k-1) * Stirling1(n,k).
Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n!, then
(2) a(n,m) = Sum_{k=0..n} m*(n+k+m)^(k-1) * Stirling1(n,k) ;
which is equivalent to the following:
(3) a(n,m) = Sum_{k=0..n} m*(n+k+m)^(k-1) * {[x^(n-k)] Product_{j=1..n-1} (1-j*x) };
(4) a(n,m) = n!*Sum_{k=0..n} m*(n+k+m)^(k-1) * {[x^(n-k)] (log(1+x)/x)^k/k!}.
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EXAMPLE
| E.g.f.: A(x) = 1 + x + 4*x^2/2! + 33*x^3/3! + 416*x^4/4! + 7100*x^5/5! +...
log(A(x)) = A(x)*log(1 + x*A(x)) where
log(A(x)) = x + 3*x^2/2! + 23*x^3/3! + 278*x^4/4! + 4624*x^5/5! + 98064*x^6/6! +...
log(1 + x*A(x)) = x + x^2/2! + 8*x^3/3! + 90*x^4/4! + 1444*x^5/5! + 29880*x^6/6! +...
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PROG
| (PARI) {a(n, m=1)=sum(k=0, n, m*(n+k+m)^(k-1)*polcoeff(prod(j=1, n-1, 1-j*x), n-k))}
(PARI) {a(n, m=1)=sum(k=0, n, m*(n+k+m)^(k-1)*n!/k!*polcoeff((log(1+x+x*O(x^n))/x)^k, n-k))}
(PARI) {Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}
{a(n, m=1)=sum(k=0, n, m*(n+k+m)^(k-1)*Stirling1(n, k))}
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CROSSREFS
| Cf. A008275 (Stirling1), A141209 (variant).
Cf. A162863. [From Paul D. Hanna (pauldhanna(AT)juno.com), Jul 19 2009]
Sequence in context: A166906 A156132 A111534 * A052885 A192548 A119821
Adjacent sequences: A162652 A162653 A162654 * A162656 A162657 A162658
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KEYWORD
| nonn
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Jul 08 2009
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