A249266 Triangle read by rows: T(n,k) is the coefficient A_k in the transformation Sum_{k=0..n} x^k = Sum_{k=0..n} A_k*(x+2*(-1)^k)^k.

1, 3, 1, -9, -3, 1, -97, -39, 7, 1, 815, 313, -65, -7, 1, 12367, 4873, -945, -127, 11, 1, -164465, -64439, 12735, 1633, -169, -11, 1, -3314673, -1302263, 255327, 33553, -3249, -263, 15, 1, 60873999, 23899401, -4695969, -613359, 60591, 4665, -321, -15, 1

Offset: 0

Author

Derek Orr, Oct 23 2014

Keywords

• sign
• tabl

Comments

Consider the transformation 1 + x + x^2 + x^3 + ... + x^n = A_0*(x+2)^0 + A_1*(x-2)^1 + A_2*(x+2)^2 + A_3*(x-2)^3 + ... + A_n*(x+2*(-1)^n)^n. This sequence gives A_0, ... A_n as the entries in the n-th row of this triangle, starting at n = 0.

Examples

			1;
3,               1;
-9,             -3,        1;
-97,           -39,        7,       1;
815,           313,      -65,      -7,     1;
12367,        4873,     -945,    -127,    11,    1;
-164465,    -64439,    12735,    1633,  -169,  -11,    1;
-3314673, -1302263,   255327,   33553, -3249, -263,   15,   1;
60873999, 23899401, -4695969, -613359, 60591, 4665, -321, -15, 1;
		

Crossrefs

Cf. A248975.

Programs

PARI

a(n,j,L)=if(j==n,return(1));if(j!=n,return(1-sum(i=1,n-j,(-L)^i*(-1)^(i*j)*binomial(i+j,i)*a(n,i+j,L))))
for(n=0,10,for(j=0,n,print1(a(n,j,2),", ")))

Formula

T(n,n-1) = 1-2*n*(-1)^n, for n > 0.