OFFSET
0,5
COMMENTS
Let F(n) denote the n-th Fibonacci number (A000045). Then n-th row sum of T is F(2n+1) and n-th alternating row sum is F(n-1).
A094436 is jointly generated with A094437 as a triangular array of coefficients of polynomials u(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, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 26 2012
This sequence gives the coefficients of the Jensen polynomials (increasing powers of x) for the sequence {A000045(k)}_{k >= 0} of degree n with shift 1. Here the definition of Jensen polynomials of degree n and shift m of an arbitrary real sequence {s(k)}_{k >= 0} is used: J(s,m;n,x) := Sum_{j=0..n} binomial(n,j)*s(j + m)*x^j, This definition is used by Griffin et al. with a different notation. - Wolfdieter Lang, Jun 25 2019
LINKS
G. C. Greubel, Rows n = 0..100 of triangle, flattened
Michael Griffin, Ken Ono, Larry Rolen, and Don Zagier, Jensen polynomials for the Riemann zeta function and other sequences, PNAS, vol. 116, no. 23, 11103-11110, June 4, 2019.
FORMULA
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(1,1) = 1 and T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Mar 26 2012
G.f. (-1+x)/(-1+2*x+x*y-x^2*y+x^2*y^2-x^2). - R. J. Mathar, Aug 11 2015
From G. C. Greubel, Oct 30 2019: (Start)
T(n, k) = binomial(n, k)*Fibonacci(k+1).
Sum_{k=0..n} T(n,k) = Fibonacci(2*n+1).
Sum_{k=0..n} (-1)^k*T(n,k) = Fibonacci(n-1). (End)
EXAMPLE
First four rows:
1
1 1
1 2 2
1 3 6 3
Sum = 1+3+6+3=13=F(7); alt.Sum = 1-3+6-3=1=F(2).
T(3,2)=F(3)C(3,2)=2*3=6.
From Philippe Deléham, Mar 26 2012: (Start)
(1, 0, 0, 1, 0, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, 0, ...) begins :
1
1, 0
1, 1, 0
1, 2, 2, 0
1, 3, 6, 3, 0
1, 4, 12, 12, 5, 0
1, 5, 20, 30, 25, 8, 0
1, 6, 30, 60, 75, 48, 13, 0 . (End)
MAPLE
with(combinat); seq(seq(fibonacci(k+1)*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+1]*Binomial[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jul 11 2019 *)
PROG
(PARI) T(n, k) = fibonacci(k+1)*binomial(n, k); \\ G. C. Greubel, Jul 11 2019
(Magma) [Fibonacci(k+1)*Binomial(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 11 2019
(Sage) [[fibonacci(k+1)*binomial(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jul 11 2019
(GAP) Flat(List([0..12], n-> List([0..n], k-> Fibonacci(k+1)* Binomial(n, k) ))); # G. C. Greubel, Jul 11 2019
CROSSREFS
KEYWORD
AUTHOR
Clark Kimberling, May 03 2004
EXTENSIONS
Offset set to 0 by Alois P. Heinz, Aug 11 2015
STATUS
approved