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A002190
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Sum_{n>=0} a(n)*x^n/n!^2 = -log(BesselJ(0,2*sqrt(x))).
(Formerly M3651 N1484)
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17
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0, 1, 1, 4, 33, 456, 9460, 274800, 10643745, 530052880, 32995478376, 2510382661920, 229195817258100, 24730000147369440, 3113066087894608560, 452168671458789789504, 75059305956331837485345, 14121026957032156557396000, 2988687741694684876495689040
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OFFSET
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0,4
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COMMENTS
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Number of non-ambiguous trees, see the Aval et al. reference. - Joerg Arndt, May 11 2015
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REFERENCES
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Stany De Smedt, On Sloane's Sequence 1484, Saitama Math. J. 15 (1997), 9-13.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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Conjecture: G.f.: 1 = Sum_{n>=0} a(n+1)*A000108(n)*x^n*Sum_{k>=0} C(2*n+k,k)^2*(-x)^k. Compare with the following g.f of the Catalan numbers (A000108): 1 = Sum_{n>=0} A000108(n)*x^n*Sum_{k>=0} C(2*n+k,k)*(-x)^k. - Paul D. Hanna, Oct 10 2010
a(n) ~ n! * (n-1)! / r^n, where r = 1/4*BesselJZero[0,1]^2 = 1.44579649073669613... - Vaclav Kotesovec, Mar 02 2014
a(0) = 0; a(n) = -(-1)^n + (1/n) * Sum_{k=1..n-1} (-1)^(n-k-1) * binomial(n,k)^2 * k * a(k). - Ilya Gutkovskiy, Jul 15 2021
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EXAMPLE
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-log( Sum_{n>=0} (-x)^n/n!^2 ) = x + x^2/2!^2 + 4*x^3/3!^2 + 33*x^4/4!^2 + 456*x^5/5!^2 + 9460*x^6/6!^2 + ... . -Paul D. Hanna, Oct 09 2010
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MAPLE
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a:= n-> coeff(series(-ln(BesselJ(0, 2*sqrt(x))), x, n+1), x, n)*(n!)^2:
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MATHEMATICA
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nn=18; CoefficientList[Series[-Log[BesselJ[0, 2*Sqrt[x]]], {x, 0, nn}], x]*Table[n!^2, {n, 0, nn}] (* Jean-François Alcover, Jun 22 2011 *)
Clear[q]; q[n_, 1] := (n-1)!^2; q[n_, k_] := q[n, k] = Sum[Binomial[n-1, j]*Binomial[n-1, j+1]*Sum[q[j+1, r]*q[n-j-1, k-r], {r, Max[1, -n+j+k+1], Min[j+1, k-1]}], { n-2}]; a[n_] := q[n, n]; Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Feb 13 2013 *)
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PROG
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(PARI)
N=66; x='x+O('x^N);
f=-log(sum(n=0, N, (-x)^n/(n!)^2) );
f=serlaplace(f);
f=serlaplace(f);
concat([0], Vec(f))
(PARI) \\ Terms starting from a(1)=1:
N=33; B=vector(N); B[1]=1; b(j)=B[j+1];
for(n=0, N-2, B[n+2]=sum(i=0, n, my(j=n-i); binomial(n+1, i)*binomial(n+1, j)*b(i)*b(j) ) ); B
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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EXTENSIONS
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Edited by Assoc. Editors of the OEIS, Oct 12 2010
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STATUS
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approved
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