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A272481 E.g.f. A(x,y) = cos((x - x*y)/2) / cos((x + x*y)/2) represented as a triangle, read by rows, where row n lists of coefficients T(n,k) of x^(2*n)*y^k/n! in A(x,y), for k=0..2*n.

%I A272481 #18 May 05 2016 21:33:16
%S A272481 1,0,1,0,0,1,3,1,0,0,3,15,25,15,3,0,0,17,119,329,455,329,119,17,0,0,
%T A272481 155,1395,5325,11235,14301,11235,5325,1395,155,0,0,2073,22803,110605,
%U A272481 311355,563013,683067,563013,311355,110605,22803,2073,0,0,38227,496951,2918825,10241231,23904881,39102063,45956625,39102063,23904881,10241231,2918825,496951,38227,0,0,929569,13943535,96075665,403075855,1150348017,2362479119,3600524785,4136759055,3600524785,2362479119,1150348017,403075855,96075665,13943535,929569,0
%N A272481 E.g.f. A(x,y) = cos((x - x*y)/2) / cos((x + x*y)/2) represented as a triangle, read by rows, where row n lists of coefficients T(n,k) of x^(2*n)*y^k/n! in A(x,y), for k=0..2*n.
%C A272481 Row sums equal the Euler numbers, A000364.
%C A272481 Column 1 equals A110501, the unsigned Genocchi numbers of first kind.
%C A272481 Main diagonal equals A272482, where A272482(n) = A005799(n)/2^n * (2*n)!/(n!)^2.
%C A272481 Sum_{k=0..2*n} (-1)^k*T(n,k) = (-1)^n.
%C A272481 Sum_{k=0..2*n} (-2)^k*T(n,k) = 2*(-1)^n for n>0.
%C A272481 Sum_{k=0..2*n} 2^k*T(n,k) = (-1)^n*A210657(n).
%C A272481 Sum_{k=0..2*n} 3^k*T(n,k) = A000281(n).
%C A272481 Sum_{k=0..2*n} 4^k*T(n,k) = A272158(n).
%C A272481 Sum_{k=0..2*n} 2^k*3^(2*n-k)*T(n,k) = A272467(n).
%H A272481 Paul D. Hanna, <a href="/A272481/b272481.txt">Table of n, a(n) for n = 0..1088, of flattened triangle for rows 0..32.</a>
%F A272481 E.g.f.: A(x,y) = (cos(x) + cos(x*y)) / (1 + cos(x + x*y)).
%F A272481 E.g.f.: A(x,y) = (sin(x) + sin(x*y)) / sin(x + x*y).
%F A272481 E.g.f.: A(x,y) = (exp(i*x) + exp(i*x*y)) / (1 + exp(i*(x + x*y))), where i^2 = -1.
%F A272481 O.g.f.: 1/(1 - 1*y*x/(1 - (1+y)^2*x/(1 - (1+2*y)*(2+1*y)*x/(1 - (2+2*y)^2*x/(1 - (2+3*y)*(3+2*y)*x/(1 - (3+3*y)^2*x/(1 - (3+4*y)*(4+3*y)*x/(1 - (4+4*y)^2*x/(1 - (4+5*y)*(5+4*y)*x/(1 - (5+5*y)^2*x/(1 - ...))))))))))), a continued fraction.
%e A272481 E.g.f.: A(x,y) = 1 + x^2*(y)/2! + x^4*(y + 3*y^2 + y^3)/4! +
%e A272481 x^6*(3*y + 15*y^2 + 25*y^3 + 15*y^4 + 3*y^5)/6! +
%e A272481 x^8*(17*y + 119*y^2 + 329*y^3 + 455*y^4 + 329*y^5 + 119*y^6 + 17*y^7)/8! +
%e A272481 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 A272481 x^12*(2073*y + 22803*y^2 + 110605*y^3 + 311355*y^4 + 563013*y^5 + 683067*y^6 + 563013*y^7 + 311355*y^8 + 110605*y^9 + 22803*y^10 + 2073*y^11)/12! +...
%e A272481 where A(x,y) = cos((x - x*y)/2) / cos((x + x*y)/2).
%e A272481 This triangle of coefficients of x^(2*n)*y^k/(2*n)!, k=0..2*n, begins:
%e A272481 [1];
%e A272481 [0, 1, 0];
%e A272481 [0, 1, 3, 1, 0];
%e A272481 [0, 3, 15, 25, 15, 3, 0];
%e A272481 [0, 17, 119, 329, 455, 329, 119, 17, 0];
%e A272481 [0, 155, 1395, 5325, 11235, 14301, 11235, 5325, 1395, 155, 0];
%e A272481 [0, 2073, 22803, 110605, 311355, 563013, 683067, 563013, 311355, 110605, 22803, 2073, 0];
%e A272481 [0, 38227, 496951, 2918825, 10241231, 23904881, 39102063, 45956625, 39102063, 23904881, 10241231, 2918825, 496951, 38227, 0];
%e A272481 [0, 929569, 13943535, 96075665, 403075855, 1150348017, 2362479119, 3600524785, 4136759055, 3600524785, 2362479119, 1150348017, 403075855, 96075665, 13943535, 929569, 0]; ...
%o A272481 (PARI) {T(n,k) = 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), k,y)}
%o A272481 for(n=0,10, for(k=0,2*n, print1(T(n,k),", "));print(""))
%Y A272481 Cf. A000364, A110501, A272482, A210657, A000281, A272158, A272467.
%K A272481 nonn,tabf
%O A272481 0,7
%A A272481 _Paul D. Hanna_, May 01 2016