A246798 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+3)^k.

1, -5, 2, 22, -16, 3, -86, 92, -33, 4, 319, -448, 237, -56, 5, -1139, 1982, -1383, 484, -85, 6, 3964, -8224, 7122, -3296, 860, -120, 7, -13532, 32600, -33702, 19384, -6700, 1392, -161, 8, 45517, -124864, 150006, -103088, 44330, -12216, 2107, -208, 9, -151313, 465626, -637314, 509272, -261850, 89844, -20573, 3032, -261, 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+3)^0 + A_1*(x+3)^1 + A_2*(x+3)^2 + ... + A_n*(x+3)^n. This sequence gives A_0, ... A_n as the entries in the n-th row of this triangle, starting at n = 0.

Examples

			Triangle starts:
1;
-5,           2;
22,         -16,       3;
-86,         92,     -33,       4;
319,       -448,     237,     -56,       5;
-1139,     1982,   -1383,     484,     -85,      6;
3964,     -8224,    7122,   -3296,     860,   -120,      7;
-13532,   32600,  -33702,   19384,   -6700,   1392,   -161,    8;
45517,  -124864,  150006, -103088,   44330, -12216,   2107, -208,    9;
-151313, 465626, -637314,  509272, -261850,  89844, -20573, 3032, -261, 10;
...
		

Crossrefs

Cf. A246788, A045944, A191008.

Programs

PARI

T(n, k) = (k+1)*sum(i=0, n-k, (-3)^i*binomial(i+k+1, k+1))
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")))

Formula

T(n,0) = (1-(4*n+5)*(-3)^(n+1))/16, for n >= 0.

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

Row n sums to (-1)^n*A045883(n+1) = T(n,0) of A246788.