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 A167769 #15 Mar 18 2021 17:46:58
%S A167769 1,1,0,0,1,1,1,0,0,1,2,3,2,1,0,0,1,3,6,8,6,3,1,0,0,1,4,10,18,24,18,10,
%T A167769 4,1,0,0,1,5,15,33,57,75,57,33,15,5,1,0,0,1,6,21,54,111,186,243,186,
%U A167769 111,54,21,6,1,0,0,1,7,28,82,193,379,622,808,622,379,193,82,28,7,1,0,0
%N A167769 Pendular trinomial triangle (p=0), read by rows of 2n+1 terms (n>=0), defined by the recurrence : if 0<k<n T(n,k)= T(n-1,k) + p*T(n,2n-1-k); else if n-1<k<2n-1, T(n,k)= T(n-1,k) + T(n,2n-2-k); with T(n,0) = T(n+1,2n) = 1 and T(n+1,2n+1) = T(n+1,2n+2) = 0.
%C A167769 See A119369 for p=1 and A122445 for p=2. The diagonals may be generated by iterated convolutions of a base sequence B (A000108(n)) with the sequence C (A000957(n+1)) of central terms.
%D A167769 Kim, Ki Hang; Rogers, Douglas G.; Roush, Fred W. Similarity relations and semiorders. Proceedings of the Tenth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1979), pp. 577--594, Congress. Numer., XXIII-XXIV, Utilitas Math., Winnipeg, Man., 1979. MR0561081 (81i:05013) - From _N. J. A. Sloane_, Jun 05 2012
%H A167769 G. C. Greubel, <a href="/A167769/b167769.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A167769 Sum_{k=0..2*n} T(n,k) = A071724(n) = [n=0] + 3*binomial(2n,n-1)/(n+2) = [n=0] + n*C(n)/(n+2), where C(n) are the Catalan numbers (A000108). - _G. C. Greubel_, Mar 17 2021
%e A167769 Triangle begins :
%e A167769 1;
%e A167769 1, 0, 0;
%e A167769 1, 1, 1, 0, 0;
%e A167769 1, 2, 3, 2, 1, 0, 0;
%e A167769 1, 3, 6, 8, 6, 3, 1, 0, 0;
%e A167769 1, 4, 10, 18, 24, 18, 10, 4, 1, 0, 0,
%e A167769 1, 5, 15, 33, 57, 75, 57, 33, 15, 5, 1, 0, 0; ...
%p A167769 T:= proc(n, k) option remember;
%p A167769 if k=0 and n=0 then 1;
%p A167769 elif k<0 or k>2*(n-1) then 0;
%p A167769 elif n=2 and k<3 then 1;
%p A167769 elif k<n then T(n, 2*n-k-1) +T(n-1, k);
%p A167769 else T(n, 2*n-k-2);
%p A167769 fi; end:
%p A167769 seq(seq(T(n, k), k=0..2*n), n=0..12); # _G. C. Greubel_, Mar 17 2021
%t A167769 T[n_, k_]:= T[n, k]= If[k==0 && n==0, 1, If[k<0 || k>2*(n-1), 0, If[n==2 && k<3, 1, If[k<n, T[n, 2*n-k-1] +T[n-1, k], T[n, 2*n-k-2] ]]]];
%t A167769 Table[T[n, k], {n, 0, 10}, {k, 0, 2*n}]//Flatten (* _G. C. Greubel_, Mar 17 2021 *)
%o A167769 (PARI) T(n, k)=if(k==0 && n==0, 1, if(k>2*n-2 || k<0, 0, if(n==2 && k<=2, 1, if(k<n, T(n-1, k)+T(n, 2*n-1-k), T(n, 2*n-2-k))))) \\ _Paul D. Hanna_, Nov 12 2009
%o A167769 (Sage)
%o A167769 @CachedFunction
%o A167769 def T(n, k):
%o A167769 if (k==0 and n==0): return 1
%o A167769 elif (k<0 or k>2*(n-1)): return 0
%o A167769 elif (n==2 and k<3): return 1
%o A167769 elif (k<n): return T(n, 2*n-k-1) +T(n-1, k)
%o A167769 else: return T(n, 2*n-k-2)
%o A167769 flatten([[T(n, k) for k in (0..2*n)] for n in (0..12)]) # _G. C. Greubel_, Mar 17 2021
%Y A167769 Cf. A000108, A000957, A000958, A001558, A001559, A071724, A104629.
%K A167769 nonn,tabf
%O A167769 0,11
%A A167769 _Philippe Deléham_, Nov 11 2009