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Search: id:A158266
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| A158266 |
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G.f.: A(x) = exp( Sum_{n>=1} C(2n-1,n)^2*x^n/n ). |
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+0
1
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| 1, 1, 5, 38, 352, 3659, 41012, 484739, 5959417, 75523708, 980470867, 12980840984, 174675568464, 2382923659387, 32890264803521, 458576476085929, 6450351908991558, 91437202854436755, 1305115286958337403
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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Compare g.f. to Catalan(x) = exp( Sum_{n>=1} C(2n-1,n)*x^n/n ), where C(2n-1,n) = A001700(n-1) and Catalan(x) is the g.f. of A000108(n) = C(2n,n)/(n+1).
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FORMULA
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a(n) = (1/n)*Sum_{k=1..n} C(2k-1,k)^2 * a(n-k) for n>0, with a(0)=1.
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EXAMPLE
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G.f.: A(x) = 1 + x + 5*x^2 + 38*x^3 + 352*x^4 + 3659*x^5 + 41012*x^6 +...
log(A(x)) = x + 3^2*x^2/2 + 10^2*x^3/3 + 35^2*x^4/4 + 126^2*x^5/5 +...
log(C(x)) = x + 3*x^2/2 + 10*x^3/3 + 35*x^4/4 + 126*x^5/5 +...
C(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 +... (g.f. of A000108).
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PROGRAM
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(PARI) {a(n)=polcoeff(exp(sum(m=1, n, binomial(2*m-1, m)^2*x^m/m)+x*O(x^n)), n)}
(PARI) {a(n)=if(n==0, 1, (1/n)*sum(k=1, n, binomial(2*k-1, k)^2*a(n-k)))}
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CROSSREFS
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Cf. A001700, A000108.
Sequence in context: A110082 A073508 A113207 this_sequence A098937 A069471 A110467
Adjacent sequences: A158263 A158264 A158265 this_sequence A158267 A158268 A158269
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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), Apr 09 2009
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