A393302 - OEIS

A393302

Numerator of Sum_{k=1..floor((n+1)/2)} binomial(n, 2*k-2) * c(k), where c(k) = A002430(k)/A036279(k) is the k-th positive coefficient in the Taylor series for tan(x).

1, 1, 2, 3, 5, 8, 587, 953, 10877, 3548, 1303712, 101437, 22983701, 26846257, 9323903014, 9151966157, 1123032621157, 204175034242, 92012223095854, 150571662913069, 117043740289223999, 829196385709459, 24137287265316486431, 3038635971927587413, 13382268659553853506859

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OFFSET

1,3

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..450

John Greene, The Burgstahler Coincidence, The Fibonacci Quarterly, Vol. 40, No. 3 (2002), pp. 194-202.

FORMULA

Let f(n) = a(n)/A393303(n). Then:

f(n) = Sum_{k=0..floor(n-1)/2} (binomial(n, 2*k) * (2/Pi)^(2*k) * Sum_{j>=0} 1/(2*j+1)^(2*k+2)).

f(n) ~ (4/Pi^2) * (1 + 2/Pi)^n.

floor(f(n)) = A000045(n) for n <= 9.

f(n)/A000045(n) ~ (4*sqrt(5)/Pi^2) * ((2+4/Pi)/(sqrt(5)+1))^n = 0.906244...*(1.011486...)^n.

The last two formulas address the "Burgstahler coincidence" (Greene, 2002), noted by the American mathematician Sylvan Burgstahler (1928-2006) at the 1999 MAA North Central Section Summer Seminar.

EXAMPLE

Fractions begin with 1, 1, 2, 3, 5, 8, 587/45, 953/45, 10877/315, 3548/63, 1303712/14175, 101437/675, ...

MATHEMATICA

c[n_] := (-1)^n * (16^n-4^n) * Zeta[1-2*n]/(2*n-1)!;

a[n_] := Numerator[Sum[Binomial[n, 2*k-2] * c[k], {k, 1, Floor[(n+1)/2]}]];

Array[a, 25]

PROG

(PARI) c(n) = (-1)^(n+1) * (16^n-4^n) * bernfrac(2*n)/(2*n)!;

a(n) = numerator(sum(k = 1, (n+1)\2, binomial(n, 2*k-2) * c(k)));

CROSSREFS

Cf. A000045, A002430, A036279, A393303 (denominators).

Sequence in context: A279074 A120495 A107477 * A271525 A357101 A232562