A248976 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, 0, 1, -3, 3, 1, 8, -6, -2, 1, 65, -50, -20, 5, 1, -296, 235, 90, -25, -4, 1, -3059, 2401, 945, -245, -49, 7, 1, 21552, -16940, -6636, 1750, 336, -56, -6, 1, 272289, -213828, -84000, 21966, 4326, -672, -90, 9, 1, -2600752, 2042805, 801996, -210126, -41160, 6570, 834, -99, -8, 1

Offset

0,4

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.

Links

Table of n, a(n) for n=0..54.

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.

Rows sum to 1.

Example

1;
0,              1;
-3,             3,      1;
8,             -6,     -2,       1;
65,           -50,    -20,       5,      1;
-296,         235,     90,     -25,     -4,    1;
-3059,       2401,    945,    -245,    -49,    7,   1;
21552,     -16940,  -6636,    1750,    336,  -56,  -6,   1;
272289,   -213828, -84000,   21966,   4326, -672, -90,   9,  1;
-2600752, 2042805, 801996, -210126, -41160, 6570, 834, -99, -8, 1

Prog

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

Crossrefs

Sequence in context: A173651 A330337 A124040 * A160332 A293294 A200342

Adjacent sequences: A248973 A248974 A248975 * A248977 A248978 A248979

Keyword

sign,tabl

Author

Derek Orr, Oct 18 2014

Status

approved