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%I #7 Jan 14 2015 15:25:03
%S 1,5,2,5,26,3,5,170,75,4,5,810,1035,164,5,5,3210,10635,3764,305,6,5,
%T 11274,91275,64244,10385,510,7,5,36362,693387,910964,261265,24030,791,
%U 8,5,110090,4822155,11361908,5422225,830430,49175,1160,9,5,317450,31364235,128935028,98319505,23510430,2226455,91880,1629,10
%N 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.
%C 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.
%F T(n,n) = n+1, n >= 0.
%F T(n,n-1) = n + 2*n^2 + 2*n^3 = A046395(n), for n >= 1.
%F T(n,n-2) = (n-1)*(2*n^4-2*n^3-2*n^2-2*n+1), for n >= 2.
%F 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.
%e From _Wolfdieter Lang_, Jan 14 2015: (Start)
%e The triangle T(n,k) starts:
%e n\k 0 1 2 3 4 5 6 7 8 9 ...
%e 0: 1
%e 1: 5
%e 2: 5 26 3
%e 3: 5 170 75 4
%e 4: 5 810 1035 164 5
%e 5: 5 3210 10635 3764 305 6
%e 6: 5 11274 91275 64244 10385 510 7
%e 7: 5 36362 693387 910964 261265 24030 791 8
%e 8: 5 110090 4822155 11361908 5422225 830430 49175 1160 9
%e 9: 5 317450 31364235 128935028 98319505 23510430 2226455 91880 1629 10
%e ... Reformatted.
%e ----------------------------------------------------------------------------
%e 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)
%o (PARI) T(n, k)=(k+1)-sum(i=k+1, n, (-2*i)^(i-k)*binomial(i, k)*T(n, i))
%o for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")))
%Y Cf. A253381, A247236, A247237.
%K nonn,tabl
%O 0,2
%A _Derek Orr_, Dec 30 2014
%E Edited. - _Wolfdieter Lang_, Jan 14 2015
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