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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.
2
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
0,2
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.
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).
EXAMPLE
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:
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
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}]
PROG
(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
KEYWORD
tabl,sign
AUTHOR
EXTENSIONS
Edited by N. J. A. Sloane, May 31 2014
STATUS
approved