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A006934 A series for Pi.
(Formerly M5119)
2
1, 1, 21, 671, 180323, 20898423, 7426362705, 1874409467055, 5099063967524835, 2246777786836681835, 2490122296790918386363, 1694873049836486741425113, 5559749161756484280905626951, 5406810236613380495234085140851, 12304442295910538475633061651918089 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Formula (21) in Luke (see ref.): Let y = 4*n+1. Then for n -> oo

Pi ~ 4*(n!)^4*2^(4*n)/(y*((2*n)!)^2)*(sum_{k>=0}((-1)^k*y^(-2*k)* A006934(k)/A123854(k)))^2. (Luke does not reference the sequences in this form.) - Peter Luschny, Mar 23 2014

This might be related to the numerators of eq. (18) in N. Elezovic' "Asymptotic Expansions of Central Binomial...", J. Int. Seq. 17 (2014) # 14.2.1. - R. J. Mathar, Mar 23 2014

Several references give an erroneous value of 1874409465055 instead of a(7) in the formula for pi. - M. F. Hasler, Mar 23 2014

REFERENCES

Y. L. Luke, The Special Functions and their Approximation, Vol. 1, Academic Press, NY, 1969, see p. 36.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Table of n, a(n) for n=0..14.

J. L. Fields, A note on the asymptotic expansion of a ratio of gamma functions, Proc. Edinburgh Math. Soc. (15), 43-45, 1966.

A. Gil, J. Segura, N. M. Temme, Fast and accurate computation of the Weber parabolic cylinder function W(a,x), IMA J. Num. Anal. 31 (2011), 1194-1216, eq (3.8).

A. Lupas, Re: Pi Calculation ?, on mathforum.org, Feb 15 2003.

C. Mortici, On some accurate estimates of pi, Bull. Math. Anal. Appl. 2(4) (2010) 137-139. (Formula (1.5), same typo as in Luke)

Index entries for sequences related to the number Pi

FORMULA

Let p(n,x) = sum(k=0..n, x^k*A220412(n,k))/A220411(n) then a(n) = (-1)^n*p(n,1/4)*A123854(n)*A001448(n). - Peter Luschny, Mar 23 2014

Pi = lim_{n->oo} 2^{4n+2}/((4n+1)*C(2n,n)^2)*(sum_{k=0..oo} (-1)^k*a(k)/(A123854(k)*(4n+1)^{2k}))^2. - M. F. Hasler, Mar 23 2014

MAPLE

A006934_list := proc(n) local k, f, bp;

bp := proc(n, x) option remember; local k; if n = 0 then 1 else -x*add(binomial(n-1, 2*k+1)*bernoulli(2*k+2)/(k+1)*bp(n-2*k-2, x), k=0..n/2-1) fi end:

f := n -> 2^(3*n-add(i, i=convert(n, base, 2)));

add(bp(2*k, 1/4)*binomial(4*k, 2*k)*x^(2*k), k=0..n-1);

seq((-1)^k*f(k)*coeff(%, x, 2*k), k=0..n-1) end:

A006934_list(15);  # Peter Luschny, Mar 23 2014

# Second solution, without using Nörlund's generalized Bernoulli polynomials, based on Euler numbers:

A006934_list := proc(n) local a, c, j;

c := n -> 4^n/2^add(i, i=convert(n, base, 2));

a := [seq((-4)^j*euler(2*j)/(4*j), j=1..n)];

expand(exp(add(a[j]*x^(-j), j=1..n))); taylor(%, x=infinity, n+2);

subs(x=1/x, convert(%, polynom)): seq(c(iquo(j, 2))*coeff(%, x, j), j=0..n) end:

A006934_list(14); # Peter Luschny, Apr 08 2014

MATHEMATICA

A006934List[n_] := Module[{c, a, s, sx}, c[k_] := 4^k/2^Total[ IntegerDigits[k, 2]]; a = Table[(-4)^j EulerE[2j]/(4j), {j, 1, n}]; s[x_] = Series[Exp[Sum[a[[j]] x^(-j), {j, 1, n}]], {x, Infinity, n+2}] // Normal; sx = s[1/x]; Table[c[Quotient[j, 2]] Coefficient[sx, x, j], {j, 0, n}]];

A006934List[14] (* Jean-François Alcover, Jun 02 2019, from second Maple program *)

PROG

(Sage)

@CachedFunction

def p(n):

    if n < 2: return 1

    return -add(binomial(n-1, k-1)*bernoulli(k)*p(n-k)/k for k in range(2, n+1, 2))/2

def A006934(n): return (-1)^n*p(2*n)*binomial(4*n, 2*n)*2^(3*n-sum(n.digits(2)))

[A006934(n) for n in (0..14)]  # Peter Luschny, Mar 24 2014

CROSSREFS

Cf. A088802, A123854, A220412.

Sequence in context: A262960 A020246 A239099 * A211877 A212734 A243746

Adjacent sequences:  A006931 A006932 A006933 * A006935 A006936 A006937

KEYWORD

nonn

AUTHOR

Simon Plouffe and N. J. A. Sloane

EXTENSIONS

a(7) corrected, a(8)-a(14) from Peter Luschny, Mar 23 2014

STATUS

approved

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Last modified January 26 11:13 EST 2020. Contains 331279 sequences. (Running on oeis4.)