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.

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, 155, 1395, 5325, 11235, 14301, 11235, 5325, 1395, 155, 0, 0, 2073, 22803, 110605, 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

Offset: 0

Author

Paul D. Hanna, May 01 2016

Keywords

• nonn
• tabf

Comments

Row sums equal the Euler numbers, A000364.

Column 1 equals A110501, the unsigned Genocchi numbers of first kind.

Main diagonal equals A272482, where A272482(n) = A005799(n)/2^n * (2*n)!/(n!)^2.

Sum_{k=0..2*n} (-1)^k*T(n,k) = (-1)^n.

Sum_{k=0..2*n} (-2)^k*T(n,k) = 2*(-1)^n for n>0.

Sum_{k=0..2*n} 2^k*T(n,k) = (-1)^n*A210657(n).

Sum_{k=0..2*n} 3^k*T(n,k) = A000281(n).

Sum_{k=0..2*n} 4^k*T(n,k) = A272158(n).

Sum_{k=0..2*n} 2^k*3^(2*n-k)*T(n,k) = A272467(n).

Examples

			E.g.f.: A(x,y) = 1 + x^2*(y)/2! + x^4*(y + 3*y^2 + y^3)/4! +
x^6*(3*y + 15*y^2 + 25*y^3 + 15*y^4 + 3*y^5)/6! +
x^8*(17*y + 119*y^2 + 329*y^3 + 455*y^4 + 329*y^5 + 119*y^6 + 17*y^7)/8! +
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! +
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! +...
where A(x,y) = cos((x - x*y)/2) / cos((x + x*y)/2).
This triangle of coefficients of x^(2*n)*y^k/(2*n)!, k=0..2*n, begins:
[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, 155, 1395, 5325, 11235, 14301, 11235, 5325, 1395, 155, 0];
[0, 2073, 22803, 110605, 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]; ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..1088, of flattened triangle for rows 0..32.

Crossrefs

Cf. A000364, A110501, A272482, A210657, A000281, A272158, A272467.

Programs

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)}
for(n=0,10, for(k=0,2*n, print1(T(n,k),", "));print(""))

Formula

E.g.f.: A(x,y) = (cos(x) + cos(x*y)) / (1 + cos(x + x*y)).

E.g.f.: A(x,y) = (sin(x) + sin(x*y)) / sin(x + x*y).

E.g.f.: A(x,y) = (exp(i*x) + exp(i*x*y)) / (1 + exp(i*(x + x*y))), where i^2 = -1.

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.