|
| |
|
|
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
|
|
|
|
FORMULA
|
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:
# 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:
|
|
|
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}]];
|
|
|
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)))
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
|
|
|
STATUS
|
approved
|
| |
|
|
