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.

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

Author

Derek Orr, Nov 15 2014

Keywords

• sign
• tabl

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.

Examples

			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;
		

Crossrefs

Cf. A248345, A045883, A014105, A001057.

Programs

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),", ")))

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).