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
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)
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
KEYWORD
nonn
AUTHOR
EXTENSIONS
a(7) corrected, a(8)-a(14) from Peter Luschny, Mar 23 2014
STATUS
approved