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E.g.f.: exp(-x*tan(x)/2).
0

%I #4 Jun 30 2022 10:39:07

%S 1,-1,-1,-3,81,5815,367791,26531589,2231306849,213451322031,

%T 21883750567455,2048495154903373,38238322254619953,

%U -84508521173113956633,-48971496478118456699569,-23204642085706390327411755,-10849535606183313563987736639,-5259600981361065001871911307681,-2687430634156750207499977165012161

%N E.g.f.: exp(-x*tan(x)/2).

%e E.g.f. A(x) = 1 - x^2/2! - x^4/4! - 3*x^6/6! + 81*x^8/8! + 5815*x^10/10! + 367791*x^12/12! + 26531589*x^14/14! + 2231306849*x^16/16! + 213451322031*x^18/18! + 21883750567455*x^20/20! + 2048495154903373*x^22/22! + 38238322254619953*x^24/24! - 84508521173113956633*x^26/26! + ...

%e The signs (+/-) of the terms a(n) begin:

%e [+,

%e -,-,-,

%e +,+,+,+,+,+,+,+,+,

%e -,-,-,-,-,-,-,-,-,-,-,-,-,

%e +,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,

%e -,-,-,-,-,-,-,-,-,-,-,-,-,-,-,-,-,-,-,-,-,-,-,

%e +,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,

%e -, ...].

%e The change in signs take place at positions:

%e [1, 4, 13, 26, 45, 68, 97, 130, 168, 211, 259, 312, 370, 433, 501, 574, 652, 734, 822, 914, 1012, 1114, 1222, 1334, 1451, 1573, 1700, 1832, 1969, 2111, 2258, 2410, 2566, 2728, 2895, ...].

%o (PARI) {a(n) = my(A = exp(-x * tan(x +O(x^(2*n+1)))/2) ); (2*n)!*polcoeff(A,2*n)}

%o for(n=0,20,print1(a(n),", "))

%Y Cf. A354245, A009273.

%K sign

%O 0,4

%A _Paul D. Hanna_, Jun 21 2022