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 A123971 #55 May 04 2025 17:23:57
%S A123971 1,2,-1,5,-5,1,13,-19,8,-1,34,-65,42,-11,1,89,-210,183,-74,14,-1,233,
%T A123971 -654,717,-394,115,-17,1,610,-1985,2622,-1825,725,-165,20,-1,1597,
%U A123971 -5911,9134,-7703,3885,-1203,224,-23,1,4181,-17345,30691,-30418,18633,-7329
%N A123971 Triangle T(n,k), read by rows, defined by T(n,k)=3*T(n-1,k)-T(n-1,k-1)-T(n-2,k), T(0,0)=1, T(1,0)=2, T(1,1)=-1, T(n,k)=0 if k<0 or if k>n.
%C A123971 This entry is the result of merging two sequences, this one and a later submission by _Philippe Deléham_, Nov 29 2013 (with edits from _Ralf Stephan_, Dec 12 2013). Most of the present version is the work of _Philippe Deléham_, the only things remaining from the original entry are the sequence data and the Mathematica program. - _N. J. A. Sloane_, May 31 2014
%C A123971 Subtriangle of the triangle given by (0, 2, 1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, -2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
%C A123971 Apart from signs, equals A126124.
%C A123971 Row sums = 1.
%C A123971 Sum_{k=0..n} T(n,k)*(-x)^k = A001519(n+1), A079935(n+1), A004253(n+1), A001653(n+1), A049685(n), A070997(n), A070998(n), A072256(n+1), A078922(n+1), A077417(n), A085260(n+1), A001570(n+1) for x=0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 respectively.
%F A123971 T(n,k) = (-1)^n*A126124(n+1,k+1).
%F A123971 T(n,k) = (-1)^k*Sum_{m=k..n} binomial(m,k)*binomial(m+n,2*m). - _Wadim Zudilin_, Jan 11 2012
%F A123971 G.f.: (1-x)/(1+(y-3)*x+x^2).
%F A123971 T(n,0) = A001519(n+1) = A000045(2*n+1).
%F A123971 T(n+1,1) = -A001870(n).
%e A123971 Triangle begins:
%e A123971 1
%e A123971 2, -1
%e A123971 5, -5, 1
%e A123971 13, -19, 8, -1
%e A123971 34, -65, 42, -11, 1
%e A123971 89, -210, 183, -74, 14, -1
%e A123971 233, -654, 717, -394, 115, -17, 1
%e A123971 Triangle (0, 2, 1/2, 1/2, 0, 0, ...) DELTA (1, -2, 0, 0, ...) begins:
%e A123971 1
%e A123971 0, 1
%e A123971 0, 2, -1
%e A123971 0, 5, -5, 1
%e A123971 0, 13, -19, 8, -1
%e A123971 0, 34, -65, 42, -11, 1
%e A123971 0, 89, -210, 183, -74, 14, -1
%e A123971 0, 233, -654, 717, -394, 115, -17, 1
%t A123971 Mathematica ( general k th center) Clear[M, T, d, a, x, k] k = 3 T[n_, m_, d_] := If[ n == m && n < d && m < d, k, If[n == m - 1 || n == m + 1, -1, If[n == m == d, k - 1, 0]]] M[d_] := Table[T[n, m, d], {n, 1, d}, {m, 1, d}] Table[M[d], {d, 1, 10}] Table[Det[M[d]], {d, 1, 10}] Table[Det[M[d] - x*IdentityMatrix[d]], {d, 1, 10}] a = Join[{M[1]}, Table[CoefficientList[ Det[M[d] - x*IdentityMatrix[d]], x], {d, 1, 10}]] Flatten[a] MatrixForm[a] Table[NSolve[Det[M[d] - x*IdentityMatrix[d]] == 0, x], {d, 1, 10}] Table[x /. NSolve[Det[M[d] - x*IdentityMatrix[d]] == 0, x][[d]], {d, 1, 10}]
%o A123971 (PARI) T(n,k)=polcoeff(polcoeff(Ser((1-x)/(1+(y-3)*x+x^2)),n,x),n-k,y) \\ _Ralf Stephan_, Dec 12 2013
%o A123971 (Sage)
%o A123971 @CachedFunction
%o A123971 def A123971(n,k): # With T(0,0) = 1!
%o A123971 if n< 0: return 0
%o A123971 if n==0: return 1 if k == 0 else 0
%o A123971 h = 2*A123971(n-1,k) if n==1 else 3*A123971(n-1,k)
%o A123971 return A123971(n-1,k-1) - A123971(n-2,k) - h
%o A123971 for n in (0..9): [A123971(n,k) for k in (0..n)] # _Peter Luschny_, Nov 20 2012
%Y A123971 Cf. A094954, A098495, A123971, A126124, A152063, A001519, A079935, A004253, A001653, A049685, A070997, A070998, A072256, A078922, A077417, A085260, A001570, A001870, A126124.
%K A123971 tabl,sign
%O A123971 0,2
%A A123971 _Gary W. Adamson_ and _Roger L. Bagula_, Oct 30 2006
%E A123971 Edited by _N. J. A. Sloane_, May 31 2014