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%I #9 May 07 2016 07:42:41
%S 1,1,3,25,455,14301,683067,45956625,4136759055,479777869285,
%T 69653708853443,12371662732650585,2638795469278520279,
%U 665531624367489426925,195911141957772574473675,66564029296445548535841825,25853886516555100370014075935,11383146182215029605579802759285,5639331535011059552351840518792275,3122923020612123544335165657648107625,1921790647828858936591348065677192750055
%N a(n) equals the coefficient of x^(2*n)*y^n/(2*n)! in cos((x - x*y)/2) / cos((x + x*y)/2).
%C Central terms of triangle A272481.
%F a(n) = A005799(n)/2^n * (2*n)!/(n!)^2 and A005799 is the generalized Euler numbers of type 2^n.
%F a(n) ~ 2^(4*n+3) * n^(2*n) / (Pi^(2*n+1) * exp(2*n)). - _Vaclav Kotesovec_, May 07 2016
%e a(n) equals the coefficient of x^(2*n)*y^n/(2*n)! in the series expansion
%e cos((x - x*y)/2) / cos((x + x*y)/2) = 1 + x^2*(y)/2! + x^4*(y + 3*y^2 + y^3)/4! +
%e x^6*(3*y + 15*y^2 + 25*y^3 + 15*y^4 + 3*y^5)/6! +
%e x^8*(17*y + 119*y^2 + 329*y^3 + 455*y^4 + 329*y^5 + 119*y^6 + 17*y^7)/8! +
%e x^10*(155*y + 1395*y^2 + 5325*y^3 + 11235*y^4 + 14301*y^5 + 11235*y^6 + 5325*y^7 + 1395*y^8 + 155*y^9)/10! +...
%e This sequence forms the central terms in triangle A272481, which begins:
%e [1];
%e [0, 1, 0];
%e [0, 1, 3, 1, 0];
%e [0, 3, 15, 25, 15, 3, 0];
%e [0, 17, 119, 329, 455, 329, 119, 17, 0];
%e [0, 155, 1395, 5325, 11235, 14301, 11235, 5325, 1395, 155, 0]; ...
%t Flatten[{1, Table[Coefficient[Series[Cos[(x - x*y)/2]/Cos[(x + x*y)/2], {x, 0, 2*n}, {y, 0, 2*n}], x^(2*n)*y^n]*(2*n)!, {n, 1, 20}]}] (* _Vaclav Kotesovec_, May 07 2016 *)
%t Table[Sum[Binomial[n, k] * Abs[EulerE[2*k]], {k, 0, n}] * Binomial[2*n,n]/4^n, {n, 0, 20}] (* faster, _Vaclav Kotesovec_, May 07 2016 *)
%o (PARI) {a(n) = my(X=x+x*O(x^(2*n))); (2*n)!*polcoeff(polcoeff( cos((X-x*y)/2)/cos((X+x*y)/2), 2*n,x), n,y)}
%o for(n=0,30, print1(a(n),", "))
%Y Cf. A272481, A005799.
%K nonn
%O 0,3
%A _Paul D. Hanna_, May 01 2016
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