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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
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
0,2
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) = 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.
EXAMPLE
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;
...
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
nonn,tabl
AUTHOR
Derek Orr, Nov 15 2014
STATUS
approved