Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #14 Oct 24 2014 04:04:06
%S 1,4,1,4,13,1,4,94,28,1,4,526,460,49,1,4,2551,5860,1399,76,1,4,11299,
%T 64180,30559,3316,109,1,4,47020,635716,566374,109156,6724,148,1,4,
%U 186988,5861188,9384358,3012196,309124,12244,193,1,4,718429,51210820,143307490,73556068,11790874,747076,20605,244,1,4,2686729,429124420,2056495090,1641197668,394515874,37488676,1608205,32644,301,1
%N 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-3k)^k for 0 <= k <= n.
%C Consider the transformation 1 + x + x^2 + x^3 + ... + x^n = A_0*(x-0)^0 + A_1*(x-3)^1 + A_2*(x-6)^2 + ... + A_n*(x-3n)^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,n-1) = 1 + 3n^2 for n > 0.
%F T(n,1) = (3^n*(n^2-n+1)-1)/2 for n > 0.
%e 1;
%e 4, 1;
%e 4, 13, 1;
%e 4, 94, 28, 1;
%e 4, 526, 460, 49, 1;
%e 4, 2551, 5860, 1399, 76, 1;
%e 4, 11299, 64180, 30559, 3316, 109, 1;
%e 4, 47020, 635716, 566374, 109156, 6724, 148, 1;
%e 4, 186988, 5861188, 9384358, 3012196, 309124, 12244, 193, 1;
%e 4, 718429, 51210820, 143307490, 73556068, 11790874, 747076, 20605, 244, 1;
%o (PARI) for(n=0, 10, for(k=0, n, if(!k, if(n, print1(4, ", ")); if(!n, print1(1, ", "))); if(k, print1(sum(i=1, n, ((3*k)^(i-k)*i*binomial(i,k)))/k, ", "))))
%Y Cf. A248977, A248830, A242598, A193843, A153703, A056107.
%K nonn,tabl
%O 0,2
%A _Derek Orr_, Oct 18 2014