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 A208153 #14 Mar 16 2025 02:34:49
%S A208153 1,1,1,3,2,1,4,7,3,1,9,14,12,4,1,14,35,31,18,5,1,28,70,87,56,25,6,1,
%T A208153 47,154,207,175,90,33,7,1,89,306,504,476,310,134,42,8,1,155,633,1137,
%U A208153 1274,941,504,189,52,9,1
%N A208153 Convolution triangle based on A006053.
%C A208153 Riordan array (1/(1-x-2*x^2+x^3), x/(1-x-2*x^2+x^3)).
%C A208153 Subtriangle of triangle given by (0, 1, 2, -5/2, 1/10, 2/5, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
%C A208153 Diagonal sums are A125691(n).
%C A208153 Row sums are A001654(n+1).
%C A208153 Mirror image of triangle in A188107.
%H A208153 Indranil Ghosh, <a href="/A208153/b208153.txt">Rows 1..100, flattened</a>
%F A208153 T(n,k) = T(n-1,k-1) + T(n-1,k) + 2*T(n-2,k) - T(n-3,k).
%F A208153 G.f.: 1/(1-x-2*x^2+x^3-y*x).
%F A208153 Sum_{k>=0} T(n-2*k,k) = A001045(n+1).
%F A208153 Sum_{k=0..n} T(n,k)*x^k = (-1)^n*A008346(n), A006053(n+2), A001654(n+1) for x = -1, 0, 1 respectively.
%e A208153 Triangle begins:
%e A208153 1
%e A208153 1, 1
%e A208153 3, 2, 1
%e A208153 4, 7, 3, 1
%e A208153 9, 14, 12, 4, 1
%e A208153 14, 35, 31, 18, 5, 1
%e A208153 Triangle (0, 1, 2, -5/2, 1/10, 2/5, 0, 0,...) DELTA (1, 0, 0, 0,...) begins:
%e A208153 1
%e A208153 0, 1
%e A208153 0, 1, 1
%e A208153 0, 3, 2, 1
%e A208153 0, 4, 7, 3, 1
%e A208153 0, 9, 14, 12, 4, 1
%e A208153 0, 14, 35, 31, 18, 5, 1
%t A208153 nmax=9; Flatten[CoefficientList[Series[CoefficientList[Series[1/(1 - x - 2*x^2 + x^3 - y*x), {x, 0, nmax}], x], {y, 0, nmax}], y]] (* _Indranil Ghosh_, Mar 10 2017 *)
%Y A208153 Cf. A006053, A000012, A000027, A055998.
%K A208153 easy,nonn,tabl
%O A208153 0,4
%A A208153 _Philippe Deléham_, Feb 24 2012