A253382 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-2k)^k.

1, 5, 2, 5, 26, 3, 5, 170, 75, 4, 5, 810, 1035, 164, 5, 5, 3210, 10635, 3764, 305, 6, 5, 11274, 91275, 64244, 10385, 510, 7, 5, 36362, 693387, 910964, 261265, 24030, 791, 8, 5, 110090, 4822155, 11361908, 5422225, 830430, 49175, 1160, 9, 5, 317450, 31364235, 128935028, 98319505, 23510430, 2226455, 91880, 1629, 10

Offset: 0

Author

Derek Orr, Dec 30 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-2)^1 + T(n,2)*(x-4)^2 + ... + T(n,n)*(x-2n)^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:  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)
		

Crossrefs

Cf. A253381, A247236, A247237.

Programs

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), ", ")))

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.

Extensions

Edited. - Wolfdieter Lang, Jan 14 2015