A248810 Signed version of A164984.

1, -1, 1, 3, -3, 1, -5, 9, -5, 1, 11, -23, 19, -7, 1, -21, 57, -61, 33, -9, 1, 43, -135, 179, -127, 51, -11, 1, -85, 313, -493, 433, -229, 73, -13, 1, 171, -711, 1299, -1359, 891, -375, 99, -15, 1, -341, 1593, -3309, 4017, -3141, 1641, -573, 129, -17, 1, 683, -3527, 8211, -11343, 10299, -6423, 2787, -831, 163, -19, 1

Offset: 0

Author

Derek Orr, Oct 14 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_n*(x+2)^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;
-1,      1;
3,      -3,     1;
-5,      9,    -5,      1;
11,    -23,    19,     -7,     1;
-21,    57,   -61,     33,    -9,     1;
43,   -135,   179,   -127,    51,   -11,    1;
-85,   313,  -493,    433,  -229,    73,  -13,    1;
171,  -711,  1299,  -1359,   891,  -375,   99,  -15,   1;
-341, 1593, -3309,   4017, -3141,  1641, -573,  129, -17,   1;
683, -3527,  8211, -11343, 10299, -6423, 2787, -831, 163, -19, 1;
		

Crossrefs

Cf. A164984, A193845, A077925, A045883.

Programs

PARI

for(n=0,20,for(k=0,n,print1(1/k!*sum(i=0,n,((-2)^(i-k)*prod(j=0,k-1,i-j))),", ")))

Formula

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

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

T(n,0) = A077925(n).

T(n,1) = (-1)^(n+1)*A045883(n).

Rows with odd n sum to 0.

Rows with even n sum to 1.