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A253384 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+3k)^k.

%I A253384 #10 Jan 22 2015 07:05:23
%S A253384 1,-5,2,-5,-34,3,-5,290,-105,4,-5,-1870,2055,-236,5,-5,10280,-30345,
%T A253384 7864,-445,6,-5,-50956,377895,-196256,22235,-750,7,-5,234812,-4194393,
%U A253384 4090264,-824485,52170,-1169,8,-5,-1024900,42834855,-75271592,25302875,-2669430,107695,-1720,9
%N A253384 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+3k)^k.
%C A253384 Consider the transformation 1 + 2x + 3x^2 + 4x^3 + ... + (n+1)*x^n = T(n,0)*(x+0)^0 + T(n,1)*(x+3)^1 + T(n,2)*(x+6)^2 + ... + T(n,n)*(x+3n)^n, for n >= 0.
%F A253384 T(n,n) = n + 1, n >= 0.
%F A253384 T(n,n-1) = n - 3*n^2 - 3*n^3, for n >= 1.
%F A253384 T(n,n-2) = (n-1)*(9*n^4 - 9*n^3 - 24*n^2 + 6*n + 2)/2, for n >= 2.
%F A253384 T(n,n-3) = (2-n)*(9*n^6 - 54*n^5 + 63*n^4 + 99*n^3 - 138*n^2 + 9*n + 10)/2, for n >= 3.
%e A253384 The triangle T(n,k) starts:
%e A253384 n\k  0       1         2        3        4      5      6    7 ...
%e A253384 0:   1
%e A253384 1:  -5       2
%e A253384 2:  -5     -34         3
%e A253384 3:  -5     290      -105        4
%e A253384 4:  -5   -1870      2055     -236        5
%e A253384 5:  -5   10280    -30345     7864     -445      6
%e A253384 6:  -5  -50956    377895  -196256    22235   -750      7
%e A253384 7:  -5  234812  -4194393  4090264  -824485  52170  -1169   8
%e A253384 ...
%e A253384 -----------------------------------------------------------------
%e A253384 n = 3: 1 + 2*x + 3*x^2 + 4*x^3 = -5*(x+0)^0 +  290*(x+3)^1 - 105*(x+6)^2 + 4*(x+9)^3.
%o A253384 (PARI) T(n, k)=(k+1)-sum(i=k+1, n, (3*i)^(i-k)*binomial(i, k)*T(n, i))
%o A253384 for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")))
%Y A253384 Cf. A247236, A247237, A253381, A253382.
%K A253384 sign,tabl
%O A253384 0,2
%A A253384 _Derek Orr_, Dec 31 2014
%E A253384 Edited; name changed, cross references added. - _Wolfdieter Lang_, Jan 22 2015