A248829 Triangle read by rows: T(n,k) is the coefficient A_k in the transformation of 1 + x + x^2 + ... + x^n to the polynomial A_k*(x+2k)^k for 0 <= k <= n .

1, -1, 1, -1, -7, 1, -1, 29, -17, 1, -1, -99, 175, -31, 1, -1, 301, -1425, 569, -49, 1, -1, -851, 10095, -8071, 1391, -71, 1, -1, 2285, -65169, 97769, -29969, 2869, -97, 1, -1, -5907, 393583, -1063447, 543471, -86731, 5279, -127, 1, -1, 14829, -2260625, 10693865, -8746257, 2181269, -212449, 8945, -161, 1, -1, -36371, 12484975, -101280535, 128879343, -48218731, 7045151, -461455, 14239, -199, 1

Offset: 0

Author

Derek Orr, Oct 15 2014

Keywords

• sign
• tabl

Comments

Consider the transformation 1 + x + x^2 + x^3 + ... + x^n = A_0*(x+0)^0 + A_1*(x+2)^1 + A_2*(x+4)^2 + ... + A_n*(x+2n)^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;
-1,    -7,        1;
-1,    29,      -17,        1;
-1,   -99,      175,      -31,        1;
-1,   301,    -1425,      569,      -49,       1;
-1,  -851,    10095,    -8071,     1391,     -71,       1;
-1,  2285,   -65169,    97769,   -29969,    2869,     -97,    1;
-1, -5907,   393583, -1063447,   543471,  -86731,    5279, -127,    1;
-1, 14829, -2260625, 10693865, -8746257, 2181269, -212449, 8945, -161, 1;
		

Crossrefs

Cf. A056220, A248826, A248830.

Programs

PARI

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

Formula

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