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 A118349 #7 Mar 17 2021 22:51:27
%S A118349 1,1,0,1,1,0,1,2,5,0,1,3,11,30,0,1,4,18,70,201,0,1,5,26,121,487,1445,
%T A118349 0,1,6,35,184,873,3592,10900,0,1,7,45,260,1375,6606,27600,85128,0,1,8,
%U A118349 56,350,2010,10672,51728,218566,682505,0,1,9,68,455,2796,15996,85182,415629,1771367,5585115,0
%N A118349 Convolution triangle, read by rows, where diagonals are successive self-convolutions of A118346.
%C A118349 A118346 equals the central terms of pendular triangle A118345 and the diagonals of this triangle form the semi-diagonals of the triangle A118345.
%H A118349 G. C. Greubel, <a href="/A118349/b118349.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A118349 Since g.f. G=G(x) of A118346 satisfies: G = 1 - 2*x*G + 2*x*G^2 + x*G^3 then T(n,k) = T(n-1,k) - 2*T(n-1,k-1) + 2*T(n,k-1) + T(n+1,k-1). Also, a recurrence involving antidiagonals is: T(n,k) = T(n-1,k) + Sum_{j=1..k} [3*T(n-1+j,k-j) - 2*T(n-2+j,k-j)] for n>k>=0.
%e A118349 Show: T(n,k) = T(n-1,k) - 2*T(n-1,k-1) + 2*T(n,k-1) + T(n+1,k-1)
%e A118349 at n=8,k=4: T(8,4) = T(7,4) - 2*T(7,3) + 2*T(8,3) + T(9,3)
%e A118349 or: 1375 = 873 - 2*184 + 2*260 + 350.
%e A118349 Triangle begins:
%e A118349 1;
%e A118349 1, 0;
%e A118349 1, 1, 0;
%e A118349 1, 2, 5, 0;
%e A118349 1, 3, 11, 30, 0;
%e A118349 1, 4, 18, 70, 201, 0;
%e A118349 1, 5, 26, 121, 487, 1445, 0;
%e A118349 1, 6, 35, 184, 873, 3592, 10900, 0;
%e A118349 1, 7, 45, 260, 1375, 6606, 27600, 85128, 0;
%e A118349 1, 8, 56, 350, 2010, 10672, 51728, 218566, 682505, 0;
%e A118349 1, 9, 68, 455, 2796, 15996, 85182, 415629, 1771367, 5585115, 0;
%p A118349 T:= proc(n, k) option remember;
%p A118349 if k<0 or k>n then 0;
%p A118349 elif k=0 then 1;
%p A118349 elif k=n then 0;
%p A118349 else T(n-1, k) -2*T(n-1, k-1) +2*T(n, k-1) +T(n+1, k-1);
%p A118349 fi; end:
%p A118349 seq(seq(T(n, k), k=0..n), n=0..12); # _G. C. Greubel_, Mar 17 2021
%t A118349 T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[k==0, 1, If[k==n, 0, T[n-1, k] -2*T[n-1, k-1] +2*T[n, k-1] +T[n+1, k-1] ]]];
%t A118349 Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* _G. C. Greubel_, Mar 17 2021 *)
%o A118349 (PARI) {T(n,k)=polcoeff((serreverse(x*(1-2*x+sqrt((1-2*x)*(1-6*x)+x*O(x^k)))/2/(1-2*x))/x)^(n-k),k)}
%o A118349 (Sage)
%o A118349 @CachedFunction
%o A118349 def T(n, k):
%o A118349 if (k<0 or k>n): return 0
%o A118349 elif (k==0): return 1
%o A118349 elif (k==n): return 0
%o A118349 else: return T(n-1, k) -2*T(n-1, k-1) +2*T(n, k-1) +T(n+1, k-1)
%o A118349 flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Mar 17 2021
%Y A118349 Cf. A118346, A118345.
%K A118349 nonn,tabl
%O A118349 0,8
%A A118349 _Paul D. Hanna_, Apr 26 2006