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%I #14 Nov 22 2014 03:54:37
%S 1,7,2,34,20,3,142,128,39,4,547,668,309,64,5,2005,3098,1929,604,95,6,
%T 7108,13304,10434,4384,1040,132,7,24604,54128,51258,27064,8600,1644,
%U 175,8,83653,211592,234966,149536,59630,15252,2443,224,9,280483,802082,1022286,761896,365810,117312,25123,3464,279,10
%N Triangle read by rows: T(n,k) is the coefficient A_k in the transformation Sum_{k=0..n} (k+1)*x^k = Sum_{k=0..n} A_k*(x-3)^k.
%C Consider the transformation 1 + 2x + 3x^2 + 4x^3 + ... + (n+1)*x^n = A_0*(x-3)^0 + A_1*(x-3)^1 + A_2*(x-3)^2 + ... + A_n*(x-3)^n. This sequence gives A_0, ... A_n as the entries in the n-th row of this triangle, starting at n = 0.
%F T(n,0) = ((2*n+1)*3^(n+1) + 1)/4, for n >= 0.
%F T(n,n-1) = n*(3*n+4), for n >= 1.
%F Row n sums to A014916(n+1) = T(2*n+1,0) of A246788.
%e Triangle starts:
%e 1;
%e 7, 2;
%e 34, 20, 3;
%e 142, 128, 39, 4;
%e 547, 668, 309, 64, 5;
%e 2005, 3098, 1929, 604, 95, 6;
%e 7108, 13304, 10434, 4384, 1040, 132, 7;
%e 24604, 54128, 51258, 27064, 8600, 1644, 175, 8;
%e 83653, 211592, 234966, 149536, 59630, 15252, 2443, 224, 9;
%e 280483, 802082, 1022286, 761896, 365810, 117312, 25123, 3464, 279, 10;
%e ...
%o (PARI) T(n, k) = (k+1)*sum(i=0, n-k, 3^i*binomial(i+k+1, k+1))
%o for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")))
%Y Cf. A246797, A014915, A140676, A246788.
%K nonn,tabl
%O 0,2
%A _Derek Orr_, Nov 15 2014
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