A246799 -id:A246799

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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.

+10
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

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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.

LINKS

Table of n, a(n) for n=0..54.

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:

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

Cf. A246788, A014106, A000337, A246799.

KEYWORD

nonn,tabl

AUTHOR

Derek Orr, Nov 15 2014

STATUS

approved

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