login
A246788
Triangle read by rows: T(n,k) is the coefficient A_k in the transformation Sum_{k=0..n} (k+1)*x^k = Sum_{k=0..n} A_k*(x+2)^k.
3
1, -3, 2, 9, -10, 3, -23, 38, -21, 4, 57, -122, 99, -36, 5, -135, 358, -381, 204, -55, 6, 313, -986, 1299, -916, 365, -78, 7, -711, 2598, -4077, 3564, -1875, 594, -105, 8, 1593, -6618, 12051, -12564, 8205, -3438, 903, -136, 9, -3527, 16422, -34029, 41196, -32115, 16722, -5817, 1304, -171, 10
OFFSET
0,2
COMMENTS
Consider the transformation 1 + 2x + 3x^2 + 4x^3 + ... + (n+1)*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.
FORMULA
T(n,0) = ((6*n+8)*(-2)^n+1)/9, for n >= 0.
T(n,n-1) = -n*(2*n+1), for n >= 1.
Row n sums to A001057(n+1).
EXAMPLE
1;
-3, 2;
9, -10, 3;
-23, 38, -21, 4;
57, -122, 99, -36, 5;
-135, 358, -381, 204, -55, 6;
313, -986, 1299, -916, 365, -78, 7;
-711, 2598, -4077, 3564, -1875, 594, -105, 8;
1593, -6618, 12051, -12564, 8205, -3438, 903, -136, 9;
-3527, 16422, -34029, 41196, -32115, 16722, -5817, 1304, -171, 10;
PROG
(PARI) T(n, k) = (k+1)*sum(i=0, n-k, (-2)^i*binomial(i+k+1, k+1))
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")))
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Derek Orr, Nov 15 2014
STATUS
approved