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A242598 Triangle read by rows: T(n,k) is the coefficient A_k in the transformation of 1 + x + x^2 + ... + x^n to the polynomial A_k*(x-k)^k for 0 <= k <= n.

%I A242598 #42 Oct 23 2014 20:48:47
%S A242598 1,2,1,2,5,1,2,14,10,1,2,30,58,17,1,2,55,258,167,26,1,2,91,978,1247,
%T A242598 386,37,1,2,140,3330,7862,4306,772,50,1,2,204,10498,44150,40146,11972,
%U A242598 1394,65,1,2,285,31234,227858,330450,153722,28610,2333,82,1,2,385,88834,1102658,2480850,1728722,482210,61133,3682,101,1
%N A242598 Triangle read by rows: T(n,k) is the coefficient A_k in the transformation of 1 + x + x^2 + ... + x^n to the polynomial A_k*(x-k)^k for 0 <= k <= n.
%C A242598 Consider the transformation 1 + x + x^2 + x^3 + ... + x^n = A_0*(x-0)^0 + A_1*(x-1)^1 + A_2*(x-2)^2 + ... + A_n*(x-n)^n. This sequence gives A_0, ... A_n as the entries in the n-th row of this triangle, starting at n = 0.
%F A242598 T(n,1) = n*(2*n+1)*(n+1)/6 for n > 0.
%F A242598 T(n,n-1) = n^2 + 1 for n > 0.
%F A242598 Rows sum to SUM{k=0..n} A138911(k).
%e A242598 1;
%e A242598 2,  1;
%e A242598 2,  5,      1;
%e A242598 2,  14,    10,       1;
%e A242598 2,  30,    58,      17,       1;
%e A242598 2,  55,   258,     167,      26,       1;
%e A242598 2,  91,   978,    1247,     386,      37,      1;
%e A242598 2, 140,  3330,    7862,    4306,     772,     50,     1;
%e A242598 2, 204, 10498,   44150,   40146,   11972,   1394,    65,    1;
%e A242598 2, 285, 31234,  227858,  330450,  153722,  28610,  2333,   82,   1;
%e A242598 2, 385, 88834, 1102658, 2480850, 1728722, 482210, 61133, 3682, 101, 1
%o A242598 (PARI) for(n=0,20,for(k=0,n,if(!k,if(n,print1(2,", "));if(!n,print1(1,", ")));if(k,print1(sum(i=1,n,(k^(i-k)*i*binomial(i,k)))/k,", "))))
%Y A242598 Cf. A000330, A002522, A138911, A248826.
%K A242598 nonn,tabl
%O A242598 0,2
%A A242598 _Derek Orr_, Oct 15 2014