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A247237
Triangle read by rows: T(n,k) is the coefficient in the transformation Sum_{k=0..n} (k+1)*x^k = Sum_{k=0..n} T(n,k)*(x-k)^k.
3
1, 3, 2, 3, 14, 3, 3, 50, 39, 4, 3, 130, 279, 84, 5, 3, 280, 1479, 984, 155, 6, 3, 532, 6519, 8544, 2675, 258, 7, 3, 924, 25335, 61464, 34035, 6138, 399, 8, 3, 1500, 89847, 388056, 356595, 106938, 12495, 584, 9, 3, 2310, 297207, 2225136, 3259635, 1524438, 284655, 23264, 819, 10
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
T(n,n) = n+1, n >= 0.
T(n,1) = n(n+1)(n+2)(3*n+1)/12 (A153978), for n >= 1.
T(n,n-1) = n^3 + n^2 + n (A027444), for n >= 1.
T(n,n-2) = (n-1)^2 (n^3-2)/2, for n >= 2.
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