A246797 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, 5, 2, 17, 14, 3, 49, 62, 27, 4, 129, 222, 147, 44, 5, 321, 702, 627, 284, 65, 6, 769, 2046, 2307, 1404, 485, 90, 7, 1793, 5630, 7683, 5884, 2725, 762, 119, 8, 4097, 14846, 23811, 22012, 12805, 4794, 1127, 152, 9, 9217, 37886, 69891, 75772, 53125, 24954, 7847, 1592, 189, 10

Offset: 0

Author

Derek Orr, Nov 15 2014

Keywords

• nonn
• 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

			Triangle starts:
1;
5,        2;
17,      14,     3;
49,      62,    27,     4;
129,    222,   147,    44,     5;
321,    702,   627,   284,    65,     6;
769,   2046,  2307,  1404,   485,    90,    7;
1793,  5630,  7683,  5884,  2725,   762,  119,    8;
4097, 14846, 23811, 22012, 12805,  4794, 1127,  152,   9;
9217, 37886, 69891, 75772, 53125, 24954, 7847, 1592, 189, 10;
...
		

Crossrefs

Cf. A246788, A014106, A000337, A246799.

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) = n*2^(n+1)+1, for n >= 0.

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

Row n sums to A014915(n-1) = T(n,0) of A246799.