A278195 Expansion of e.g.f. (1/6!)*sin^6(x)/cos(x) (coefficients of even powers only).

0, 0, 0, 1, -28, 882, -17116, 803803, 13713336, 3671012164, 506128123928, 96524822605365, 21542790273363260, 5676618945053498806, 1739246268204447115932, 613255488134158250903887, 246554708506039690689322544, 112115693433705109495581088008

Offset

0,5

Comments

This sequence gives the coefficients in an asymptotic expansion of a series related to the constant Pi. It can be shown that (1/6!)*Pi/4 = Sum_{k >= 1} (-1)^k/((2*k - 7)*(2*k - 5)*(2*k - 3)*(2*k - 1)*(2*k + 1)*(2*k + 3)*(2*k + 5)). Using Proposition 1 of Borwein et al. it can be shown that the following asymptotic expansion holds for the tails of this series: for N divisible by 4, 2*{ (1/6!)*Pi/4 - Sum_{k = 1..N/2} (-1)^k/((2*k - 7)*(2*k - 5)*(2*k - 3)*(2*k - 1)*(2*k + 1)*(2*k + 3)*(2*k + 5)) } ~ -1/N^7 + (-28)/N^9 - 882/N^11 + (-17116)/N^13 - 803803/N^15 + .... An example is given below.

Links

Table of n, a(n) for n=0..17.

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)!] ( 1/6!*sin^6(x)/cos(x) ).

a(n) = (1/6!)*( A000364(n) - 1/16*((-25)^n - 7*(-9)^n + 22*(-1)^n) ).

a(n) = (-1)^(n+1)/(2^6*6!) * 2^(2*n)*( E(2*n,7/2) - 6*E(2*n,5/2) + 15*E(2*n,3/2) - 20*E(2*n,1/2) + 15*E(2*n,-1/2) - 6*E(2*n,-3/2) + E(2*n,-5/2) ), where E(n,x) is the Euler polynomial of order n.

Example

Expansion of (1/6!)*sin^6(x)/cos(x) starts x^6/6! - 28*x^8/8! + 882*x^10/10! - 17116*x^12/12! + ....
Let N = 100000. The truncated series 2*Sum_{k = 1..N/2} (-1)^k/( (2*k - 7)*(2*k - 5)*(2*k - 3)*(2*k - 1)*(2*k + 1)*(2*k + 3)*(2*k + 5) ) = 0.0021816615649929119711546134606107(7)5891803(617)860677(2450)20121.... The bracketed digits show where this decimal expansion differs from that of Pi/1440. The numbers -1, -28, -882 must be added to the bracketed numbers to give the correct decimal expansion: Pi/1440 = 0.0021816615649929119711546134606107(6)5891803(589)860677(1568)20121....

Maple

A000364 := n -> abs(euler(2*n)):
seq((1/6!)*(A000364(n) - (1/16)*((-25)^n - 7*(-9)^n + 22*(-1)^n) ), n = 0..20);

Mathematica

With[{nn=40}, Take[CoefficientList[Series[Sin[x]^6/Cos[x] 1/6!, {x, 0, nn}], x] Range[0, nn]!, {1, -1, 2}]] (* Harvey P. Dale, Sep 12 2019 *)

Crossrefs

Cf. A000364, A004174, A166984, A278079, A278080, A278194.

Sequence in context: A119055 A324450 A326279 * A095656 A057412 A160263

Adjacent sequences: A278192 A278193 A278194 * A278196 A278197 A278198

Keyword

sign,easy

Author

Peter Bala, Nov 15 2016

Status

approved