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A247237 Triangle read by rows: T(n,k) is the coefficient in the transformation Sum_{k=0..n} (k+1)*x^k = Sum_{k=0..n} T(n,k)*(x-k)^k.

%I A247237 #18 Apr 26 2016 12:31:53
%S A247237 1,3,2,3,14,3,3,50,39,4,3,130,279,84,5,3,280,1479,984,155,6,3,532,
%T A247237 6519,8544,2675,258,7,3,924,25335,61464,34035,6138,399,8,3,1500,89847,
%U A247237 388056,356595,106938,12495,584,9,3,2310,297207,2225136,3259635,1524438,284655,23264,819,10
%N A247237 Triangle read by rows: T(n,k) is the coefficient in the transformation Sum_{k=0..n} (k+1)*x^k = Sum_{k=0..n} T(n,k)*(x-k)^k.
%C A247237 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.
%F A247237 T(n,n) = n+1, n >= 0.
%F A247237 T(n,1) = n(n+1)(n+2)(3*n+1)/12 (A153978), for n >= 1.
%F A247237 T(n,n-1) = n^3 + n^2 + n (A027444), for n >= 1.
%F A247237 T(n,n-2) = (n-1)^2 (n^3-2)/2, for n >= 2.
%e A247237 From _Wolfdieter Lang_, Jan 14 2015: (Start)
%e A247237 The triangle T(n,k) starts:
%e A247237 n\k 0    1      2       3       4       5      6     7   8  9 ...
%e A247237 0:  1
%e A247237 1:  3    2
%e A247237 2:  3   14      3
%e A247237 3:  3   50     39       4
%e A247237 4:  3  130    279      84       5
%e A247237 5:  3  280   1479     984     155       6
%e A247237 6:  3  532   6519    8544    2675     258      7
%e A247237 7:  3  924  25335   61464   34035    6138    399     8
%e A247237 8:  3 1500  89847  388056  356595  106938  12495   584   9
%e A247237 9:  3 2310 297207 2225136 3259635 1524438 284655 23264 819 10
%e A247237 ...
%e A247237 -----------------------------------------------------------------
%e A247237 n = 3: 1 + 2*x + 3*x^2 + 4*x^3 = 3*(x-0)^0 +  50*(x-1)^1 + 39*(x-2)^2 + 4*(x-3)^3.
%e A247237 (End)
%o A247237 (PARI) T(n,k)=(k+1)-sum(i=k+1,n,(-i)^(i-k)*binomial(i,k)*T(n,i))
%o A247237 for(n=0,10,for(k=0,n,print1(T(n,k),", ")))
%Y A247237 Cf. A247236, A153978, A027444, A253381.
%K A247237 nonn,tabl
%O A247237 0,2
%A A247237 _Derek Orr_, Nov 27 2014
%E A247237 Edited by _Wolfdieter Lang_, Jan 14 2015