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A024199
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a(0) = 0, a(1) = 1, a(n+1) = 2*a(n) + (2*n-1)^2*a(n-1).
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13
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0, 1, 2, 13, 76, 789, 7734, 110937, 1528920, 28018665, 497895210, 11110528485, 241792844580, 6361055257725, 163842638377950, 4964894559637425, 147721447995130800, 5066706567801827025, 171002070002301095250, 6548719685561840296125, 247199273204273879989500
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| (2*n+1)!!/a(n+1), n>=0, is the n-th approximant for William Brouncker's continued fraction for 4/Pi =1+1^2/(2+3^2/(2+5^2/(2+... See the C. Brezinski and J.-P. Delahaye references given under A142969 and A142970, respectively. The double factorials (2*n+1)!! = A001147(n+1) enter. W. Lang, Oct 06 2008. [From Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Oct 06 2008]
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REFERENCES
| A. E. Jolliffe, Continued Fractions, in Encyclopaedia Britannica, 11th ed., pp. 30-33; see p. 31.
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FORMULA
| a(n) = s(1)s(2)...s(n)(1/s(1) - 1/s(2) + ... + c/s(n)) where c=(-1)^(n+1) and s(k) = 2k-1 for k = 1, 2, 3, ...
A024199(n) + A024200(n) = A001147(n) = (2n-1)!! - Max Alekseyev (maxale(AT)gmail.com), Sep 23 2007.
A024199(n)/A024200(n) -> Pi/(4-Pi) as n -> oo. - Max Alekseyev (maxale(AT)gmail.com), Sep 23 2007.
Contribution from Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Oct 06 2008: (Start)
E.g.f. for a(n+1), n>=0: (sqrt(1-2*x)+arcsin(2*x)*sqrt(1+2*x)/2)/((1-4*x^2)^(1/2)*(1-2*x)). From the recurrence, solving (1-4*x^2)y''(x)-2*(8*x+1)*y'(x)-9*y=0 with inputs y(0)=1, y'(0)=2.
a(n+1)= A003148(n) + A143165(n), n>=0 (from the two terms of the e.g.f.). (End)
Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Nov 12 2009: (Start)
a(n) = (-1)^(n-1)*(2*n-3)!! + (2*n-1)*a(n-1) with a(0) = 0.
a(n) = (2*n-1)!!*sum((-1)^(k)/(2*k+1), k=0..n-1)
(End)
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MAPLE
| f := proc(n) option remember; local a, b, t1, t2, t3, i, j, k; a := 0; b := 1; if n=0 then RETURN(a) elif n=1 then RETURN(b) else RETURN(2*f(n-1)+ (2*n-3)^2*f(n-2)); fi; end;
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MATHEMATICA
| f[k_] := (2 k - 1) (-1)^(k + 1)
t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[n - 1, t[n]]
Table[a[n], {n, 1, 22}] (* A024199 signed *)
(* Clark Kimberling, Dec 30 2011 *)
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CROSSREFS
| Cf. A004041.
Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Nov 12 2009: (Start)
Cf. A007509 and A025547.
Equals first column of A167584.
Equals row sums of A167591.
Equals first right hand column of A167594.
(End)
Sequence in context: A192700 A007509 A077413 * A037523 A037732 A090187
Adjacent sequences: A024196 A024197 A024198 * A024200 A024201 A024202
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KEYWORD
| nonn
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AUTHOR
| Clark Kimberling (ck6(AT)evansville.edu)
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EXTENSIONS
| Edited by N. J. A. Sloane (njas(AT)research.att.com), Jul 19 2002.
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