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%I #20 Mar 01 2023 11:00:43
%S 0,0,1,-5,126,1490,118151,8256885,808428076,100199284180,
%T 15432169163901,2889536106161375,646438926423519626,
%U 170294687860735726470,52177485058722877649251,18397662218707151323777465,7396641315814156362154666776
%N Expansion of e.g.f. (1/4!)*sin^4(x)/cos(x) (coefficients of even powers only).
%C 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.
%H J. M. Borwein, P. B. Borwein, K. Dilcher, <a href="http://www.jstor.org/stable/2324715">Pi, Euler numbers and asymptotic expansions</a>, Amer. Math. Monthly, 96 (1989), 681-687.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EulerPolynomial.html">Euler Polynomial</a>.
%F a(n) = [x^(2*n)/(2*n)!] ( (1/4!)*sin^4(x)/cos(x) ).
%F a(n) = (1/4!)*( A000364(n) + (-1)^n*(9^(n) - 5)/4 ).
%F 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.
%F 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! + ....
%F 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 - ....
%e 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.. ..
%p A000364 := n -> abs(euler(2*n)):
%p seq(1/4!*(A000364(n) + (-1)^n*(9^n - 5)/4), n = 0..20);
%Y Cf. A000364, A004174, A024235, A166984, A278079, A278194, A278195.
%K sign,easy
%O 0,4
%A _Peter Bala_, Nov 10 2016
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