A248975 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+(-1)^k)^k for 0 <= k <= n.

1, 2, 1, -1, -1, 1, -12, -10, 4, 1, 45, 34, -14, -3, 1, 406, 319, -124, -33, 6, 1, -2357, -1847, 731, 187, -39, -5, 1, -26968, -21188, 8312, 2182, -424, -68, 8, 1, 223769, 175700, -69052, -18034, 3566, 548, -76, -7, 1, 3096810, 2432333, -955048, -250126, 49052, 7730, -1000, -115, 10, 1

Offset: 0

Author

Derek Orr, Oct 18 2014

Keywords

• sign
• tabl

Comments

Consider the transformation 1 + x + x^2 + x^3 + ... + x^n = A_0*(x+1)^0 + A_1*(x-1)^1 + A_2*(x+1)^2 + A_3*(x-1)^3 + ... + A_n*(x+(-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;
2,             1;
-1,           -1,       1;
-12,         -10,       4,       1;
45,           34,     -14,      -3,     1;
406,         319,    -124,     -33,     6,    1;
-2357,     -1847,     731,     187,   -39,   -5,     1;
-26968,   -21188,    8312,    2182,  -424,  -68,     8,    1;
223769,   175700,  -69052,  -18034,  3566,  548,   -76,   -7,  1;
3096810, 2432333, -955048, -250126, 49052, 7730, -1000, -115, 10, 1
		

Programs

PARI

a(n,j) = if(j==n,return(1));if(j!=n,return(1-sum(i=1,n-j,(-1)^(i*(j+1))*binomial(i+j,i)*a(n,i+j))))
for(n=0,15,for(j=0,n,print1(a(n,j),", ")))

Formula

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

T(n,n-2) = (1-n)*((3/2)*n-(-1)^n) + 1 for n > 1.

T(n,0) = 1 - sum(i=1..n) (-1)^i*T(n,i) = 1 + T(n,1) - T(n-2) + T(n-3) - ... + (-1)^(n-1)*T(n,n-1) + (-1)^n.