%I #51 Aug 09 2015 01:18:00

%S 1,2,-1,5,-5,1,13,-19,8,-1,34,-65,42,-11,1,89,-210,183,-74,14,-1,233,

%T -654,717,-394,115,-17,1,610,-1985,2622,-1825,725,-165,20,-1,1597,

%U -5911,9134,-7703,3885,-1203,224,-23,1,4181,-17345,30691,-30418,18633,-7329

%N 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 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 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 Apart from signs, equals A126124.

%C Row sums = 1.

%C 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 T(n,k)= (-1)^n*A126124(n+1,k+1).

%F T(n,k) = (-1)^k*sum(m=k,n, binomial(m,k)*binomial(m+n,2*m) ). - _Wadim Zudilin_, Jan 11 2012

%F G.f.: (1-x)/(1+(y-3)*x+x^2).

%F T(n,0) = A001519(n+1) = A000045(2*n+1).

%F T(n+1,1) = -A001870(n).

%e Triangle begins:

%e 1

%e 2, -1

%e 5, -5, 1

%e 13, -19, 8, -1

%e 34, -65, 42, -11, 1

%e 89, -210, 183, -74, 14, -1

%e 233, -654, 717, -394, 115, -17, 1

%e Triangle (0, 2, 1/2, 1/2, 0, 0, ...) DELTA (1, -2, 0, 0, ...) begins:

%e Triangle (0, 2, 1/2, 1/2, 0, 0, ...) DELTA (1, -2, 0, 0, ...) begins:

%e 1

%e 0, 1

%e 0, 2, -1

%e 0, 5, -5, 1

%e 0, 13, -19, 8, -1

%e 0, 34, -65, 42, -11, 1

%e 0, 89, -210, 183, -74, 14, -1

%e 0, 233, -654, 717, -394, 115, -17, 1

%t 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 (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 (Sage)

%o @CachedFunction

%o def A123971(n,k): # With T(0,0) = 1!

%o if n< 0: return 0

%o if n==0: return 1 if k == 0 else 0

%o h = 2*A123971(n-1,k) if n==1 else 3*A123971(n-1,k)

%o return A123971(n-1,k-1) - A123971(n-2,k) - h

%o for n in (0..9): [A123971(n,k) for k in (0..n)] # _Peter Luschny_, Nov 20 2012

%Y Cf. A094954, A098495, A123971, A126124, A152063, A001519, A079935, A004253, A001653, A049685, A070997, A070998, A072256, A078922, A077417, A085260, A001570, A001870, A126124.

%K tabl,sign

%O 0,2

%A _Gary W. Adamson_ and _Roger L. Bagula_, Oct 30 2006

%E Edited by _N. J. A. Sloane_, May 31 2014