OFFSET
0,3
COMMENTS
Let F(n) denote the n-th Fibonacci number (A000045). Then n-th row sum of T is F(2n+2) and n-th alternating row sum is -F(n-2).
A094437 is jointly generated with A094436 as a triangular array of coefficients of polynomials v(n,x): initially, u(1,x)=v(1,x)=1; for n>1, u(n,x)=u(n-1,x)+x*v(n-1)x and v(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x). See the Mathematica section. [Clark Kimberling, Feb 26 2012]
Subtriangle of the triangle given by (1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Apr 28 2012
LINKS
G. C. Greubel, Rows n = 0..100 of triangle, flattened
FORMULA
From Philippe Deléham, Apr 28 2012: (Start)
As DELTA-triangle T(n,k):
G.f.: (1-x-y*x+2*y*x^2-y^2*x^2)/(1-2*x-y*x+x^2+y*x^2-y^2*x^2).
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k) - T(n-2,k-1) + T(n-2,k-2), T(0,0) = T(1,0) = T(2,0) = 1, T(2,1) = 2, T(1,1) = T(2,2) = 0 and T(n,k) = 0 if k<0 or if k>n. (End)
From G. C. Greubel, Oct 30 2019: (Start)
T(n, k) = binomial(n, k)*Fibonacci(k+2).
Sum_{k=0..n} T(n,k) = Fibonacci(2*n+2).
Sum_{k=0..n} (-1)^(k+1) * T(n,k) = Fibonacci(n-2). (End)
EXAMPLE
First four rows:
1;
1 2;
1 4 3;
1 6 9 5;
sum = 1+6+9+5=21=F(8); alt.sum = 1-6+9-5=-1=-F(1).
T(3,2)=F(4)*C(3,2)=3*3=9.
From Philippe Deléham, Apr 28 2012: (Start)
(1, 0, 0, 1, 0, 0, ...) DELTA (0, 2, -1/2, -1/2, 0, 0, ...) begins :
1;
1, 0;
1, 2, 0;
1, 4, 3, 0;
1, 6, 9, 5, 0;
1, 8, 18, 20, 8, 0; . (End)
MAPLE
with(combinat); seq(seq(fibonacci(k+2)*binomial(n, k), k=0..n), n=0..12); # G. C. Greubel, Oct 30 2019
MATHEMATICA
(* First program *)
u[1, x_] := 1; v[1, x_] := 1; z = 13;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x];
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%] (* A094436 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A094437 *)
(* Second program *)
Table[Fibonacci[k+2]*Binomial[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Oct 30 2019 *)
PROG
(PARI) T(n, k) = binomial(n, k)*fibonacci(k+2);
for(n=0, 12, for(k=0, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Oct 30 2019
(Magma) [Binomial(n, k)*Fibonacci(k+2): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 30 2019
(Sage) [[binomial(n, k)*fibonacci(k+2) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Oct 30 2019
(GAP) Flat(List([0..12], n-> List([0..n], k-> Binomial(n, k)*Fibonacci(k+2) ))); # G. C. Greubel, Oct 30 2019
CROSSREFS
KEYWORD
AUTHOR
Clark Kimberling, May 03 2004
STATUS
approved