A247236 Triangle read by rows: T(n,k) appears in the transformation Sum_{k=0..n} (k+1)*x^k = Sum_{k=0..n} T(n,k)*(x+k)^k.

1, -1, 2, -1, -10, 3, -1, 26, -33, 4, -1, -54, 207, -76, 5, -1, 96, -993, 824, -145, 6, -1, -156, 4047, -6736, 2375, -246, 7, -1, 236, -14769, 46184, -28985, 5634, -385, 8, -1, -340, 49743, -280408, 293575, -95166, 11711, -568, 9, -1, 470, -157617, 1556672, -2609465, 1322334, -260449, 22112, -801, 10

Offset: 0

Author

Derek Orr, Nov 27 2014

Keywords

• sign
• 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 12 2015: (Start)
The triangle T(n,k) starts:
n\k 0    1       2       3        4       5       6     7    8  9 ...
0:  1
1: -1    2
2: -1  -10       3
3: -1   26     -33       4
4: -1  -54     207     -76        5
5: -1   96    -993     824     -145       6
6: -1 -156    4047   -6736     2375    -246       7
7: -1  236  -14769   46184   -28985    5634    -385     8
8: -1 -340   49743 -280408   293575  -95166   11711  -568    9
9: -1  470 -157617 1556672 -2609465 1322334 -260449 22112 -801 10
... Reformatted.
---------------------------------------------------------------------
n=3: 1 + 2*x + 3*x^2 + 4*x^3 = -1*(x+0)^0 + 26*(x+1)^1 - 33*(x+2)^2 + 4*(x+3)^3. (End)
		

Crossrefs

Cf. A248345, A000027, A002717, A085490.

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 = A000027(n+1), n >= 0.

T(n,1) = ((-1)^n*(1-4*n^3-10*n^2-4*n)-1)/8 = 2*(-1)^(n+1)*A002717(n), for n >= 1.

T(n,n-1) = n - n^2 - n^3 (A085490), for n >= 1.

T(n,n-2) = (n^5-2*n^4-3*n^3+6*n^2-2)/2, for n >= 2.

Extensions

Edited. - Wolfdieter Lang, Jan 12 2015