A248978 - 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