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-1)^1 + T(n,2)*(x-2)^2 + ... + T(n,n)*(x-n)^n, for n >= 0.
FORMULA
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: 3 2
2: 3 14 3
3: 3 50 39 4
4: 3 130 279 84 5
5: 3 280 1479 984 155 6
6: 3 532 6519 8544 2675 258 7
7: 3 924 25335 61464 34035 6138 399 8
8: 3 1500 89847 388056 356595 106938 12495 584 9
9: 3 2310 297207 2225136 3259635 1524438 284655 23264 819 10
...
-----------------------------------------------------------------
n = 3: 1 + 2*x + 3*x^2 + 4*x^3 = 3*(x-0)^0 + 50*(x-1)^1 + 39*(x-2)^2 + 4*(x-3)^3.
(End)
PROG
(PARI) T(n, k)=(k+1)-sum(i=k+1, n, (-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, Nov 27 2014
EXTENSIONS
Edited by Wolfdieter Lang, Jan 14 2015
STATUS
approved