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A247236 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+k)^k.

%I A247236 #14 Jan 12 2015 17:50:08
%S A247236 1,-1,2,-1,-10,3,-1,26,-33,4,-1,-54,207,-76,5,-1,96,-993,824,-145,6,
%T A247236 -1,-156,4047,-6736,2375,-246,7,-1,236,-14769,46184,-28985,5634,-385,
%U A247236 8,-1,-340,49743,-280408,293575,-95166,11711,-568,9,-1,470,-157617,1556672,-2609465,1322334,-260449,22112,-801,10
%N A247236 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+k)^k.
%C A247236 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 A247236 T(n,n) = n+1 = A000027(n+1), n >= 0.
%F A247236 T(n,1) = ((-1)^n*(1-4*n^3-10*n^2-4*n)-1)/8 = 2*(-1)^(n+1)*A002717(n), for n >= 1.
%F A247236 T(n,n-1) = n - n^2 - n^3 (A085490), for n >= 1.
%F A247236 T(n,n-2) = (n^5-2*n^4-3*n^3+6*n^2-2)/2, for n >= 2.
%e A247236 From _Wolfdieter Lang_, Jan 12 2015: (Start)
%e A247236 The triangle T(n,k) starts:
%e A247236 n\k 0    1       2       3        4       5       6     7    8  9 ...
%e A247236 0:  1
%e A247236 1: -1    2
%e A247236 2: -1  -10       3
%e A247236 3: -1   26     -33       4
%e A247236 4: -1  -54     207     -76        5
%e A247236 5: -1   96    -993     824     -145       6
%e A247236 6: -1 -156    4047   -6736     2375    -246       7
%e A247236 7: -1  236  -14769   46184   -28985    5634    -385     8
%e A247236 8: -1 -340   49743 -280408   293575  -95166   11711  -568    9
%e A247236 9: -1  470 -157617 1556672 -2609465 1322334 -260449 22112 -801 10
%e A247236 ... Reformatted.
%e A247236 ---------------------------------------------------------------------
%e A247236 n=3: 1 + 2*x + 3*x^2 + 4*x^3 = -1*(x+0)^0 + 26*(x+1)^1 - 33*(x+2)^2 + 4*(x+3)^3. (End)
%o A247236 (PARI) T(n,k)=(k+1)-sum(i=k+1,n,i^(i-k)*binomial(i,k)*T(n,i))
%o A247236 for(n=0,10,for(k=0,n,print1(T(n,k),", ")))
%Y A247236 Cf. A248345, A000027, A002717, A085490.
%K A247236 sign,tabl
%O A247236 0,3
%A A247236 _Derek Orr_, Nov 27 2014
%E A247236 Edited. - _Wolfdieter Lang_, Jan 12 2015