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Search: id:A161629
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| A161629 |
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E.g.f. satisfies: A(x) = exp( x*Catalan(x*A(x)) ), where Catalan(x) = (1-sqrt(1-4*x))/(2*x) is the g.f. of A000108. |
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1
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| 1, 1, 3, 25, 349, 6821, 171421, 5265625, 191160201, 8007548617, 380157603481, 20171371753061, 1182973489103869, 75984447924612397, 5305029326492409333, 400014338565211619761, 32396515980658185762961
(list; graph; listen)
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OFFSET
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0,3
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FORMULA
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E.g.f.: A(x) = Sum_{n>=0} a(n)*x^n/n!, where
a(n) = Sum_{k=0..n} n! * (n-k+1)^(k-1)/k! * C(2*n-k,n-k)*k/(2*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(2*n-k,n-k)*k/(2*n-k).
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EXAMPLE
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E.g.f.: A(x) = 1 + x + 3*x^2/2! + 25*x^3/3! + 349*x^4/4! + 6821*x^5/5! +...
log(A(x))/x = 1 + x*A(x) + 2*x^2*A(x)^2 + 5*x^3*A(x)^3 + 14*x^4*A(x)^4 +...+ A000108(n)*x^n*A(x)^n +...
log(A(x))/x = 1 + x + 6*x^2/2! + 63*x^3/3! + 988*x^4/4! + 20725*x^5/5! +...
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PROGRAM
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(PARI) {a(n, m=1)=if(n==0, 1, sum(k=0, n, n!/k!*m*(m+n-k)^(k-1)*binomial(2*n-k, n-k)*k/(2*n-k)))}
(PARI) {a(n, m=1)=local(A=1+x+x*O(x^n)); for(i=1, n, A=exp((1-sqrt(1-4*x*A))/(2*A))); n!*polcoeff(A^m, n)}
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CROSSREFS
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Cf. A000108.
Sequence in context: A154961 A085527 A093360 this_sequence A129506 A143139 A012481
Adjacent sequences: A161626 A161627 A161628 this_sequence A161630 A161631 A161632
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KEYWORD
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nonn
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AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), Jun 17 2009
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