A253384 Triangle read by rows: T(n,k) appears in the transformation Sum_{k=0..n} (k+1)*x^k = Sum_{k=0..n} T(n,k)*(x+3k)^k.

1, -5, 2, -5, -34, 3, -5, 290, -105, 4, -5, -1870, 2055, -236, 5, -5, 10280, -30345, 7864, -445, 6, -5, -50956, 377895, -196256, 22235, -750, 7, -5, 234812, -4194393, 4090264, -824485, 52170, -1169, 8, -5, -1024900, 42834855, -75271592, 25302875, -2669430, 107695, -1720, 9

Offset: 0

Author

Derek Orr, Dec 31 2014

Keywords

• sign
• tabl

Comments

Consider the transformation 1 + 2x + 3x^2 + 4x^3 + ... + (n+1)*x^n = T(n,0)*(x+0)^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 ...
0:   1
1:  -5       2
2:  -5     -34         3
3:  -5     290      -105        4
4:  -5   -1870      2055     -236        5
5:  -5   10280    -30345     7864     -445      6
6:  -5  -50956    377895  -196256    22235   -750      7
7:  -5  234812  -4194393  4090264  -824485  52170  -1169   8
...
-----------------------------------------------------------------
n = 3: 1 + 2*x + 3*x^2 + 4*x^3 = -5*(x+0)^0 +  290*(x+3)^1 - 105*(x+6)^2 + 4*(x+9)^3.
		

Crossrefs

Cf. A247236, A247237, A253381, A253382.

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) = n + 1, n >= 0.

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 - 24*n^2 + 6*n + 2)/2, for n >= 2.

T(n,n-3) = (2-n)*(9*n^6 - 54*n^5 + 63*n^4 + 99*n^3 - 138*n^2 + 9*n + 10)/2, for n >= 3.

Extensions

Edited; name changed, cross references added. - Wolfdieter Lang, Jan 22 2015