A162169 Exponential series expansion of (cos(x) - sin(x))*cosh(t*x) + sinh(t*x).

1, -1, 1, -1, 0, 1, 1, 0, -3, 1, 1, 0, -6, 0, 1, -1, 0, 10, 0, -5, 1, -1, 0, 15, 0, -15, 0, 1, 1, 0, -21, 0, 35, 0, -7, 1, 1, 0, -28, 0, 70, 0, -28, 0, 1, -1, 0, 36, 0, -126, 0, 84, 0, -9, 1, -1, 0, 45, 0, -210, 0, 210, 0, -45, 0, 1, 1, 0, -55, 0, 330, 0, -462, 0, 165, 0, -11, 1

Offset: 1

Author

Mats Granvik, Jun 27 2009

Keywords

• sign
• tabl

Comments

Previous name was: Signed version of Pascal's triangle.

Related to A000111 via its matrix inverse A162170.

For odd columns k, T(n, k) = binomial(n-1, k-1) * (-1)^floor((n+k-1)/2). For even columns, T(n, k) = 1 if n = k, otherwise 0. - Mike Tryczak, Jun 17 2015

From Peter Bala, Sep 08 2021: (Start)

In the notation of the Bala link, this is the array [[ cos(x) - sin(x), 1 ]] with inverse array A162170 = [[ sec(x) + tan(x), 1 ]].

In general, arrays of the form [[ G(x), 1 ]], where G(x) = 1 + g(1)*x + g(2)*x^2/2! + g(3)*x^3/3! + ... is an e.g.f., form a group with group law [[ G(x), 1 ]]*[[ F(x), 1 ]] = [[ G(x)*F_e(x) + F_o(x), 1 ]] and inverse array [[ G(x), 1 ]]^(-1) = [[ (1 - G_o(x))/G_e(x), 1 ]], where G_e(x) = (G(x) + G(-x))/2 and G_o(x) = (G(x) - G(-x))/2 are the even and odd parts of G(x). (End)

Examples

			Table begins:
   1;
  -1,    1;
  -1,    0,    1;
   1,    0,   -3,    1;
   1,    0,   -6,    0,    1;
  -1,    0,   10,    0,   -5,    1;
  -1,    0,   15,    0,  -15,    0,    1;
   1,    0,  -21,    0,   35,    0,   -7,    1;
   1,    0,  -28,    0,   70,    0,  -28,    0,    1;
  -1,    0,   36,    0, -126,    0,   84,    0,   -9,    1;
  -1,    0,   45,    0, -210,    0,  210,    0,  -45,    0,    1;
   1,    0,  -55,    0,  330,    0, -462,    0,  165,    0,  -11,    1;
.
As a symmetric triangle:
                                   1;
                               -1,    1;
                            -1,    0,    1;
                          1,    0,   -3,    1;
                       1,    0,   -6,    0,    1;
                   -1,    0,   10,    0,   -5,    1;
                -1,    0,   15,    0,  -15,    0,    1;
              1,    0,  -21,    0,   35,    0,   -7,    1;
           1,    0,  -28,    0,   70,    0,  -28,    0,    1;
       -1,    0,   36,    0, -126,    0,   84,    0,   -9,    1;
    -1,    0,   45,    0, -210,    0,  210,    0,  -45,    0,    1;
  1,    0,  -55,    0,  330,    0, -462,    0,  165,    0,  -11,    1;
		

Links

• Peter Bala, Matrices with repeated columns - the generalised Appell groups

Crossrefs

Cf. A007318, A162170.

Programs

Excel

=if(or(mod(row()-column();4)=1;mod(row()-column();4)=2);-1;1)*if(row()>=column();combin(row()-1;column()-1);0)*if(and(row()>column();mod(column();2)=0);0;1)

Maple

egf := (cos(x) - sin(x))*cosh(t*x) + sinh(t*x):
ser := n -> series(egf, x, n+1): c := n -> n!*coeff(ser(n), x, n):
A162169row := n -> seq(coeff(c(n), t, k), k=0..n):
for n from 0 to 9 do A162169row(n) od; # Peter Luschny, Sep 18 2021

Mathematica

nn=12; Flatten[Table[Table[If[Or[Mod[n - k, 4] == 1, Mod[n - k, 4] == 2], -1, 1]*If[n >= k, Binomial[n - 1, k - 1], 0]*If[And[n > k, Mod[k, 2] == 0], 0, 1], {k, 1, n}], {n, 1, nn}]] (* Mats Granvik, Nov 25 2017 *)

PARI

T(n, k) = if (k % 2, binomial(n-1, k-1) * (-1)^floor((n+k-1)/2), if (n==k, 1 , 0));
tabl(nn) = {for (n=1, nn, for (k=1, n, print1(T(n,k), ", ");); print(););} \\ Michel Marcus, Jun 17 2015

Formula

E.g.f.: (cos(x) - sin(x))*cosh(t*x) + sinh(t*x) = 1 + (-1 + t)*x + (-1 + t^2)*x^2/2! + (1 - 3^t^2 + t^3)*x^3/3! + .... - Peter Bala, Sep 08 2021

Extensions

New name using a formula of Peter Bala from Peter Luschny, Sep 18 2021