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#18 by Vaclav Kotesovec at Thu Mar 06 12:19:09 EST 2014
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#17 by Vaclav Kotesovec at Thu Mar 06 12:18:59 EST 2014
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| FORMULA
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Conjecture: Limit n->infinity a(n)^(1/n^2) = 4. - Vaclav Kotesovec, Mar 06 2014
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#16 by Vaclav Kotesovec at Thu Mar 06 12:12:33 EST 2014
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| MATHEMATICA
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nmax = 10; b = ConstantArray[0, nmax+1]; b[[1]] = 1; Do[b[[n+1]] = 1/n*Sum[Binomial[2*k, k]^k/2^k * b[[n-k+1]], {k, 1, n}], {n, 1, nmax}]; b (* Vaclav Kotesovec, Mar 06 2014 *)
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| STATUS
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approved
editing
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#15 by Paul D. Hanna at Wed Apr 17 00:58:23 EDT 2013
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#14 by Paul D. Hanna at Wed Apr 17 00:58:19 EDT 2013
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| NAME
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G.f.: exp( Sum_{n>=1} C(2*n,n)^n/2^n * x^n/n ).
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| COMMENTS
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Compare to the g.f. C(x) = 1 + x*C(x)^2 of the Catalan numbers (A000108), where C(x) = exp( Sum_{n>=1} C(2*n,n)/2 * x^n/n ).
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| EXAMPLE
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G.f.: A(x) = 1 + x + 5*x^2 + 338*x^3 + 375502*x^4 + 6351970709*x^5 +...
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| PROG
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((PARI) {a(n)=polcoeff(exp(sum(m=1, n, binomial(2*m, m)^m/2^m*x^m/m)+x*O(x^n)), n)}
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| CROSSREFS
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Cf. A001700A224732, A201556, A001700.
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| STATUS
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approved
editing
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#13 by Russ Cox at Fri Mar 30 18:37:32 EDT 2012
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| AUTHOR
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_Paul D. Hanna (pauldhanna(AT)juno.com), _, Dec 05 2011
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Discussion
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| Fri Mar 30
| 18:37
| OEIS Server: https://oeis.org/edit/global/213
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#12 by Paul D. Hanna at Mon Dec 05 13:15:21 EST 2011
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| NAME
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G.f.: exp( Sum_{n>=1} C(2*n,n)^n/2^n * x^n/n ).
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| COMMENTS
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Compare to the g.f. C(x) = 1 + x*C(x)^2 of the Catalan numbers (A000108), where C(x) = exp( Sum_{n>=1} C(2*n,n)/2 * x^n/n ).
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| EXAMPLE
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G.f.: A(x) = 1 + x + 5*x^2 + 338*x^3 + 375502*x^4 + 6351970709*x^5 +...
where the logarithm of the g.f. begins:
where
Compare to the logarithm of the Catalan function C(x) = (1-sqrt(1-4*x))/(2*x):
log(C(x)) = x + 3*x^2/2 + 10*x^3/3 + 35*x^4/4 + 126*x^5/5 + 462*x^6/6 +...
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| PROG
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( (PARI) {a(n)=polcoeff(exp(sum(m=1, n, binomial(2*m, m)^m/2^m*x^m/m)+x*O(x^n)), n)}
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| CROSSREFS
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Cf. A001700, A201556.
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| STATUS
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proposed
approved
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#11 by Paul D. Hanna at Mon Dec 05 13:14:25 EST 2011
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#10 by Paul D. Hanna at Mon Dec 05 13:14:20 EST 2011
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| NAME
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G.f.: exp( Sum_{n>=1} C(2*n,n)^n/2^n * x^n/n ).
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| COMMENTS
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Compare to the g.f. C(x) = 1 + x*C(x)^2 of the Catalan numbers (A000108), where C(x) = exp( Sum_{n>=1} C(2*n,n)/2 * x^n/n ).
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| EXAMPLE
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G.f.: A(x) = 1 + x + 5*x^2 + 338*x^3 + 375502*x^4 + 6351970709*x^5 +...
where the logarithm of the g.f. begins:
Compare to the logarithm of the Catalan function C(x) = (1-sqrt(1-4*x))/(2*x):
log(C(x)) = x + 3*x^2/2 + 10*x^3/3 + 35*x^4/4 + 126*x^5/5 + 462*x^6/6 +...
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| PROG
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((PARI) {a(n)=polcoeff(exp(sum(m=1, n, binomial(2*m, m)^m/2^m*x^m/m)+x*O(x^n)), n)}
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| CROSSREFS
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Cf. A001700, A201556.
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| KEYWORD
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nonn,newchanged
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| STATUS
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approved
editing
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#9 by T. D. Noe at Mon Dec 05 13:14:01 EST 2011
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