A006934 A series for Pi.

1, 1, 21, 671, 180323, 20898423, 7426362705, 1874409467055, 5099063967524835, 2246777786836681835, 2490122296790918386363, 1694873049836486741425113, 5559749161756484280905626951, 5406810236613380495234085140851, 12304442295910538475633061651918089

Offset: 0

Author

Simon Plouffe and N. J. A. Sloane

Keywords

• nonn

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

• 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

Crossrefs

Cf. A088802, A123854, A220412.

Programs

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 *)

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

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

Extensions

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