A135220 Matrices C_n (read by rows), defined by the following identity: C_n * P(X,Y) = P(X+Y, X-Y), where P is any homogeneous polynomial of degree n in two variables, represented as a column whose i-th element is the coefficient of X^(n-i)Y^i.

1, 1, 1, 1, -1, 1, 1, 1, 2, 0, -2, 1, -1, 1, 1, 1, 1, 1, 3, 1, -1, -3, 3, -1, -1, 3, 1, -1, 1, -1, 1, 1, 1, 1, 1, 4, 2, 0, -2, -4, 6, 0, -2, 0, 6, 4, -2, 0, 2, -4, 1, -1, 1, -1, 1, 1, 1, 1, 1, 1, 1, 5, 3, 1, -1, -3, -5, 10, 2, -2, -2, 2, 10, 10, -2, -2, 2, 2, -10, 5, -3, 1, 1, -3, 5, 1, -1, 1, -1, 1, -1

Offset

1,9

Links

Table of n, a(n) for n=1..91.

Formula

Formula: (C_n)ij = Sum(k=0..j, (-1)^k * C(j, k) * C(n-j, i-k)).

Recurrence: (C_n)i,j = (C_(n-1))i,j + (C_(n-1))i-1,j for all j < n; (C_n)i,j = (C_(n-1))i,j-1 - (C_(n-1))i-1,j-1 for all j > 0.

Example

C_0 = [1]
C_1 = [1 1 / 1 -1]
C_2 is equal to:
|1 1 1|
|2 0 -2|
|1 -1 1|
since if
P(X,Y) = a0*X^2 + a1*XY + a2*Y^2
then
P(X+Y,X-Y) = (a0 + a1 + a2)*X^2 + (2*a0 - 2*a2)*XY + (a0 - a1 + a2)*Y^2

Crossrefs

Sequence in context: A094451 A056562 A280008 * A347550 A068320 A111330

Adjacent sequences: A135217 A135218 A135219 * A135221 A135222 A135223

Keyword

easy,tabf,sign

Author

Ilia Smilga (ilia.smilga(AT)orange.fr), Feb 14 2008, Feb 20 2008

Status

approved