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A024199 a(0) = 0, a(1) = 1, a(n+1) = 2*a(n) + (2*n-1)^2*a(n-1). 13
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; text; internal format)
OFFSET

0,3

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, Oct 06 2008]

REFERENCES

A. E. Jolliffe, Continued Fractions, in Encyclopaedia Britannica, 11th ed., pp. 30-33; see p. 31.

LINKS

Table of n, a(n) for n=0..20.

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, Sep 23 2007.

A024199(n)/A024200(n) -> Pi/(4-Pi) as n -> oo. - Max Alekseyev, Sep 23 2007.

Contribution from Wolfdieter Lang, 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, 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)

E.g.f.: Pi/4/sqrt(1-2*x) - 1/2*log(2*x+sqrt(4*x^2-1))/sqrt(2*x-1). - Vaclav Kotesovec, Mar 18 2014

a(n) ~ Pi * 2^(n-3/2) * n^n / exp(n). - Vaclav Kotesovec, Mar 18 2014

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;

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 *)

RecurrenceTable[{a[n+1] == 2*a[n] + (2*n-1)^2*a[n-1], a[0] == 0, a[1] == 1}, a, {n, 0, 20}] (* Vaclav Kotesovec, Mar 18 2014 *)

CoefficientList[Series[Pi/4/Sqrt[1-2*x] - 1/2*Log[2*x+Sqrt[4*x^2-1]]/Sqrt[2*x-1], {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Mar 18 2014 *)

CROSSREFS

Cf. A004041.

Contribution from Johannes W. Meijer, 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

KEYWORD

nonn

AUTHOR

Clark Kimberling

EXTENSIONS

Edited by N. J. A. Sloane, Jul 19 2002.

STATUS

approved

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Last modified November 25 23:19 EST 2014. Contains 250017 sequences.