A278080 Expansion of e.g.f. (1/4!)*sin^4(x)/cos(x) (coefficients of even powers only).

0, 0, 1, -5, 126, 1490, 118151, 8256885, 808428076, 100199284180, 15432169163901, 2889536106161375, 646438926423519626, 170294687860735726470, 52177485058722877649251, 18397662218707151323777465, 7396641315814156362154666776

Offset

0,4

Comments

This sequence gives the coefficients in an asymptotic expansion of a series related to the constant Pi. It can be shown that (1/4!)*Pi/4 = Sum_{k >= 1} (-1)^(k-1)/((2*k - 5)*(2*k - 3)*(2*k - 1)*(2*k + 1)*(2*k + 3)). Using Proposition 1 of Borwein et al. it can be shown that the following asymptotic expansion holds for the tails of the series: for N divisible by 4, 2*( (1/4!)*Pi/4 - Sum_{k = 1..N/2} (-1)^(k-1)/((2*k - 5)*(2*k - 3)*(2*k - 1)*(2*k + 1)*(2*k + 3)) ) ~ 1/N^5 - (-5)/N^7 + 126/N^9 - 1490/N^11 + 118151/N^13 - .... An example is given below. Cf. A024235 and A278195.

Links

Table of n, a(n) for n=0..16.

J. M. Borwein, P. B. Borwein, K. Dilcher, Pi, Euler numbers and asymptotic expansions, Amer. Math. Monthly, 96 (1989), 681-687.

Eric Weisstein's World of Mathematics, Euler Polynomial.

Formula

a(n) = [x^(2*n)/(2*n)!] ( (1/4!)*sin^4(x)/cos(x) ).

a(n) = (1/4!)*( A000364(n) + (-1)^n*(9^(n) - 5)/4 ).

a(n) = (-1)^n/(2^4*4!) * 2^(2*n)*( E(2*n,5/2) - 4*E(2*n,3/2) + 6*E(2*n,1/2) - 4*E(2*n,-1/2) + E(2*n,-3/2) ), where E(n,x) is the Euler polynomial of order n.

E.g.f. (1/4!)*sin^4(x)/cos(x) = x^4/4! - 5*x^6/6! + 126*x^8/8! + 1490*x^10/10! + ....

O.g.f. for a signed version of the sequence: Sum_{n >= 0} ( (1/2^n) * Sum_{k = 0..n} (-1)^k*binomial(n, k)/((1 - (2*k - 3)*x)*(1 - (2*k - 1)*x)*(1 - (2*k + 1)*x)*(1 - (2*k + 3)*x)*(1 - (2*k + 5)*x)) ) = 1 + 5*x^2 + 126*x^4 - 1490*x^6 + 118151*x^8 - ....

Example

Let N = 100000. The truncated series 2*Sum_{k = 1..N/2} (-1)^(k-1)/((2*k - 5)*(2*k - 3)*(2*k - 1)*(2*k + 1)*(2*k + 3)) = 0.065449846949787359134638(3)038183229(2)6754107(569)820314(8536)0364.... The bracketed digits show where this decimal expansion differs from that of Pi/48. The numbers 1, 5, 126, -1490 must be added to the bracketed numbers to give the correct decimal expansion to 60 digits: Pi/48 = 0.065449846949787359134638(4) 038183229(7)6754107(695)820314(7046)0364.. ..

Maple

A000364 := n -> abs(euler(2*n)):
seq(1/4!*(A000364(n) + (-1)^n*(9^n - 5)/4), n = 0..20);

Crossrefs

Cf. A000364, A004174, A024235, A166984, A278079, A278194, A278195.

Sequence in context: A229868 A201839 A234609 * A156956 A015476 A059486

Adjacent sequences: A278077 A278078 A278079 * A278081 A278082 A278083

Keyword

sign,easy

Author

Peter Bala, Nov 10 2016

Status

approved