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-2)^1 + T(n,2)*(x-4)^2 + ... + T(n,n)*(x-2n)^n, for n >= 0.
FORMULA
T(n,n) = n+1, n >= 0.
T(n,n-1) = n + 2*n^2 + 2*n^3 = A046395(n), for n >= 1.
T(n,n-2) = (n-1)*(2*n^4-2*n^3-2*n^2-2*n+1), for n >= 2.
T(n,n-3) = (n-2)*(4*n^6-24*n^5+38*n^4-6*n^3+12*n^2-36*n+15)/3, for n >= 3.
EXAMPLE
From Wolfdieter Lang, Jan 14 2015: (Start)
The triangle T(n,k) starts:
n\k 0 1 2 3 4 5 6 7 8 9 ...
0: 1
1: 5
2: 5 26 3
3: 5 170 75 4
4: 5 810 1035 164 5
5: 5 3210 10635 3764 305 6
6: 5 11274 91275 64244 10385 510 7
7: 5 36362 693387 910964 261265 24030 791 8
8: 5 110090 4822155 11361908 5422225 830430 49175 1160 9
9: 5 317450 31364235 128935028 98319505 23510430 2226455 91880 1629 10
... Reformatted.
----------------------------------------------------------------------------
n = 3: 1 + 2*x + 3*x^2 + 4*x^3 = 5*(x-0)^0 + 170*(x-2)^1 + 75*(x-4)^2 + 4*(x-6)^3. (End)
PROG
(PARI) T(n, k)=(k+1)-sum(i=k+1, n, (-2*i)^(i-k)*binomial(i, k)*T(n, i))
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")))
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Derek Orr, Dec 30 2014
EXTENSIONS
Edited. - Wolfdieter Lang, Jan 14 2015
STATUS
approved