A253383 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-3k)^k.

1, 7, 2, 7, 38, 3, 7, 362, 111, 4, 7, 2522, 2271, 244, 5, 7, 14672, 34671, 8344, 455, 6, 7, 75908, 442911, 212464, 23135, 762, 7, 7, 361676, 5015199, 4498984, 869855, 53682, 1183, 8, 7, 1621388, 52044447, 83860840, 26997215, 2775282, 110047, 1736, 9, 7, 6935798, 505540767, 1423092160, 732435935, 117592782, 7458367, 205856, 2439, 10

Offset: 0

Author

Derek Orr, Dec 30 2014

Keywords

• nonn
• tabl

Comments

Consider the transformation 1 + 2x + 3x^2 + 4x^3 + ... + (n+1)*x^n = T(n,0)*(x-3)^0 + T(n,1)*(x-3)^1 + T(n,2)*(x-6)^2 + ... + T(n,n)*(x-3n)^n, for n >= 0.

Examples

			The triangle T(n,k) starts:
n\k 0        1         2         3         4        5       6     7  8  ...
0:  1
1:  7        2
2:  7       38         3
3:  7      362       111         4
4:  7     2522      2271       244         5
5:  7    14672     34671      8344       455        6
6:  7    75908    442911    212464     23135      762       7
7:  7   361676   5015199   4498984    869855    53682    1183     8
8:  7  1621388  52044447  83860840  26997215  2775282  110047  1736  9
...
-----------------------------------------------------------------
n = 3: 1 + 2*x + 3*x^2 + 4*x^3 = 7*(x-0)^0 +  362*(x-3)^1 + 111*(x-6)^2 + 4*(x-9)^3.
		

Crossrefs

Cf. A253382, A247237.

Programs

PARI

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

Formula

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

T(n,n-2) = (n-1)*(9*n^4 - 9*n^3 - 12*n^2 - 6*n + 2)/2, for n >= 2.

T(n,n-3) = (n-2)*(9*n^6 - 54*n^5 + 81*n^4 + 9*n^3 - 12*n^2 - 45*n + 14)/2, for n >= 3.