A094436 Triangular array T(n,k) = Fibonacci(k+1)*binomial(n,k) for k = 0..n; n >= 0.

1, 1, 1, 1, 2, 2, 1, 3, 6, 3, 1, 4, 12, 12, 5, 1, 5, 20, 30, 25, 8, 1, 6, 30, 60, 75, 48, 13, 1, 7, 42, 105, 175, 168, 91, 21, 1, 8, 56, 168, 350, 448, 364, 168, 34, 1, 9, 72, 252, 630, 1008, 1092, 756, 306, 55, 1, 10, 90, 360, 1050, 2016, 2730, 2520, 1530, 550, 89

Offset: 0

Author

Clark Kimberling, May 03 2004

Keywords

• nonn
• easy
• tabl

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

Examples

			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)
		

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.

Crossrefs

Cf. A000045.

Cf. A094435, A094437, A094438, A094439, A094440, A094441, A094442, A094443, A094444.

Programs

GAP

Flat(List([0..12], n-> List([0..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

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 *)

PARI

T(n,k) = fibonacci(k+1)*binomial(n,k); \\ 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

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)

Extensions

Offset set to 0 by Alois P. Heinz, Aug 11 2015