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A253381 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.

%I A253381 #6 Jan 12 2015 17:17:22
%S A253381 1,-3,2,-3,-22,3,-3,122,-69,4,-3,-518,891,-156,5,-3,1882,-8709,3444,
%T A253381 -295,6,-3,-6182,71931,-57036,9785,-498,7,-3,18906,-530181,789684,
%U A253381 -241095,23022,-777,8,-3,-54822,3598587,-9661260,4919865,-783378,47607,-1144,9,-3,152538,-22943493,107911860,-87977415,21896622,-2129673,89576,-1611,10
%N A253381 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 A253381 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 A253381 T(n,n)   = n+1, for n >= 0.
%F A253381 T(n,n-1) = n*(1 - 2*n - 2*n^2), for n >= 1.
%F A253381 T(n,n-2) = (n-1)*(2*n^4-2*n^3-6*n^2+2*n+1), for n >= 2.
%F A253381 T(n,n-3) = (2-n)*(4*n^6-24*n^5+26*n^4+54*n^3-72*n^2+9)/3, for n >= 3.
%e A253381 From - _Wolfdieter Lang_, Jan 12 2015: (Start)
%e A253381 The triangle T(n,k) starts:
%e A253381 n\k  0      1         2         3         4        5        6     7     8  9 ...
%e A253381 0:   1
%e A253381 1:  -3      2
%e A253381 2:  -3    -22         3
%e A253381 3:  -3    122       -69         4
%e A253381 4:  -3   -518       891      -156         5
%e A253381 5:  -3   1882     -8709      3444      -295        6
%e A253381 6:  -3  -6182     71931    -57036      9785     -498        7
%e A253381 7:  -3  18906   -530181    789684   -241095    23022     -777     8
%e A253381 8:  -3 -54822   3598587  -9661260   4919865  -783378    47607 -1144     9
%e A253381 9 : -3 152538 -22943493 107911860 -87977415 21896622 -2129673 89576 -1611 10
%e A253381 ... Reformatted.
%e A253381 ----------------------------------------------------------------------------------
%e A253381 n = 3: 1 + 2*x + 3*x^2 + 4*x^3 = -3*(x+0)^0 + 122*(x+2)^1 - 69*(x+4)^2 + 4* (x+6)^3. (End)
%o A253381 (PARI) T(n,k) = (k+1)-sum(i=k+1,n,(2*i)^(i-k)*binomial(i,k)*T(n,i))
%o A253381 for(n=0,10,for(k=0,n,print1(T(n,k),", ")))
%Y A253381 Cf. A247236.
%K A253381 sign,tabl
%O A253381 0,2
%A A253381 _Derek Orr_, Dec 30 2014
%E A253381 Edited; - _Wolfdieter Lang_, Jan 12 2015