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.

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

Author

Derek Orr, Nov 27 2014

Keywords

• nonn
• 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-1)^1 + T(n,2)*(x-2)^2 + ... + T(n,n)*(x-n)^n, for n >= 0.

Examples

			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)
		

Crossrefs

Cf. A247236, A153978, A027444, A253381.

Programs

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),", ")))

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.

Extensions

Edited by Wolfdieter Lang, Jan 14 2015