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.

1, 2, -1, 5, -5, 1, 13, -19, 8, -1, 34, -65, 42, -11, 1, 89, -210, 183, -74, 14, -1, 233, -654, 717, -394, 115, -17, 1, 610, -1985, 2622, -1825, 725, -165, 20, -1, 1597, -5911, 9134, -7703, 3885, -1203, 224, -23, 1, 4181, -17345, 30691, -30418, 18633, -7329

Offset: 0

Author

Gary W. Adamson and Roger L. Bagula, Oct 30 2006

Keywords

• tabl
• sign

Comments

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

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.

Apart from signs, equals A126124.

Row sums = 1.

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.

Examples

			Triangle begins:
  1
  2, -1
  5, -5, 1
  13, -19, 8, -1
  34, -65, 42, -11, 1
  89, -210, 183, -74, 14, -1
  233, -654, 717, -394, 115, -17, 1
Triangle (0, 2, 1/2, 1/2, 0, 0, ...) DELTA (1, -2, 0, 0, ...) begins:
  1
  0, 1
  0, 2, -1
  0, 5, -5, 1
  0, 13, -19, 8, -1
  0, 34, -65, 42, -11, 1
  0, 89, -210, 183, -74, 14, -1
  0, 233, -654, 717, -394, 115, -17, 1
		

Crossrefs

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

Programs

Mathematica

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}]

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

Sage

@CachedFunction
def A123971(n,k): # With T(0,0) = 1!
    if n< 0: return 0
    if n==0: return 1 if k == 0 else 0
    h = 2*A123971(n-1,k) if n==1 else 3*A123971(n-1,k)
    return A123971(n-1,k-1) - A123971(n-2,k) - h
for n in (0..9): [A123971(n,k) for k in (0..n)] # Peter Luschny, Nov 20 2012

Formula

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

T(n,k) = (-1)^k*Sum_{m=k..n} binomial(m,k)*binomial(m+n,2*m). - Wadim Zudilin, Jan 11 2012

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

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

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

Extensions

Edited by N. J. A. Sloane, May 31 2014