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A002067
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a(n) = Sum_{k=0..n-1} binomial(2*n,2*k)*a(k)*a(n-k-1).
(Formerly M4458 N1889)
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7
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1, 1, 7, 127, 4369, 243649, 20036983, 2280356863, 343141433761, 65967241200001, 15773461423793767, 4591227123230945407, 1598351733247609852849, 655782249799531714375489, 313160404864973852338669783, 172201668512657346455126457343, 108026349476762041127839800617281
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OFFSET
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0,3
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COMMENTS
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Also number of increasing rooted triangular cacti of 2n+1 nodes. (In an increasing rooted graph, nodes are numbered and numbers increase as you move away from root.)
a(n) is (2n)!/2^n times the n-th coefficient in the series for InverseErf[2x/Sqrt[Pi]]. [From Paul Barry, Apr 12 2010]
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REFERENCES
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Cf. Chapter 5 of F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998.
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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T. D. Noe, Table of n, a(n) for n=0..50
Wikipedia, Error Function
Index entries for sequences related to cacti
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FORMULA
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We have a(n)=b(2n+1), where e.g.f. of b satisfies B'(x)=exp(B(x)^2/2).
a(n)=(2n)!*A092676(n)/(2^n*A092677(n)). [From Paul Barry, Apr 12 2010]
a(n) = 1/2^n*A026944(n+1). Let D denote the operator g(x) -> 1/sqrt(2)*d/dx(exp(x^2)*g(x)). Then a(n) = D^(2*n)(1) evaluated at x = 0. - Peter Bala, Sep 08 2011
E.g.f. A(x) satisfies differential equation A''(x)-A(x)*A''(x)-1=0, A'(0)=1/2. [From Vladimir Kruchinin, Nov 18 2011]
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MAPLE
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a:=proc(n) option remember; if n <= 0 then RETURN(1); else RETURN( add( binomial(2*n, 2*k)*a(k)*a(n-k-1), k=0..n-1 ) ); fi; end;
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MATHEMATICA
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max = 16; se = Series[ InverseErf[ 2*x/Sqrt[Pi] ], {x, 0, 2*max+1} ]; a[n_] := (2n+1)!/2^n*Coefficient[ se, x, 2*n+1]; Table[ a[n], {n, 0, max} ](* From Jean-François Alcover, Mar 07 2012, after Paul Barry *)
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CROSSREFS
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The sequence of fractions A092676/A132467 is closely related.
Periods: A122149, A122159.
Sequence in context: A139291 A215066 A092676 * A138523 A034670 A020516
Adjacent sequences: A002064 A002065 A002066 * A002068 A002069 A002070
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KEYWORD
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nonn,eigen,easy,nice
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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Alternate description, formula and comment from Christian G. Bower.
New definition and more terms from Vladeta Jovovic, Oct 22 2005
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STATUS
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approved
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