This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A253383 #9 Jan 16 2015 09:19:34
%S A253383 1,7,2,7,38,3,7,362,111,4,7,2522,2271,244,5,7,14672,34671,8344,455,6,
%T A253383 7,75908,442911,212464,23135,762,7,7,361676,5015199,4498984,869855,
%U A253383 53682,1183,8,7,1621388,52044447,83860840,26997215,2775282,110047,1736,9,7,6935798,505540767,1423092160,732435935,117592782,7458367,205856,2439,10
%N A253383 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-3k)^k.
%C A253383 Consider the transformation 1 + 2x + 3x^2 + 4x^3 + ... + (n+1)*x^n = T(n,0)*(x-3)^0 + T(n,1)*(x-3)^1 + T(n,2)*(x-6)^2 + ... + T(n,n)*(x-3n)^n, for n >= 0.
%F A253383 T(n,n-1) = n + 3*n^2 + 3*n^3, for n >= 1.
%F A253383 T(n,n-2) = (n-1)*(9*n^4 - 9*n^3 - 12*n^2 - 6*n + 2)/2, for n >= 2.
%F A253383 T(n,n-3) = (n-2)*(9*n^6 - 54*n^5 + 81*n^4 + 9*n^3 - 12*n^2 - 45*n + 14)/2, for n >= 3.
%e A253383 The triangle T(n,k) starts:
%e A253383 n\k 0 1 2 3 4 5 6 7 8 ...
%e A253383 0: 1
%e A253383 1: 7 2
%e A253383 2: 7 38 3
%e A253383 3: 7 362 111 4
%e A253383 4: 7 2522 2271 244 5
%e A253383 5: 7 14672 34671 8344 455 6
%e A253383 6: 7 75908 442911 212464 23135 762 7
%e A253383 7: 7 361676 5015199 4498984 869855 53682 1183 8
%e A253383 8: 7 1621388 52044447 83860840 26997215 2775282 110047 1736 9
%e A253383 ...
%e A253383 -----------------------------------------------------------------
%e A253383 n = 3: 1 + 2*x + 3*x^2 + 4*x^3 = 7*(x-0)^0 + 362*(x-3)^1 + 111*(x-6)^2 + 4*(x-9)^3.
%o A253383 (PARI) T(n, k)=(k+1)-sum(i=k+1, n, (-3*i)^(i-k)*binomial(i, k)*T(n, i))
%o A253383 for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")))
%Y A253383 Cf. A253382, A247237.
%K A253383 nonn,tabl
%O A253383 0,2
%A A253383 _Derek Orr_, Dec 30 2014