OFFSET
0,2
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.
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.
EXAMPLE
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.
PROG
(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), ", ")))
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Derek Orr, Dec 31 2014
EXTENSIONS
Edited; name changed, cross references added. - Wolfdieter Lang, Jan 22 2015
STATUS
approved