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Triangular array T(n,k) = Fibonacci(k+1)*binomial(n,k) for k = 0..n; n >= 0.
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%I #44 Sep 08 2022 08:45:13

%S 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,

%T 7,42,105,175,168,91,21,1,8,56,168,350,448,364,168,34,1,9,72,252,630,

%U 1008,1092,756,306,55,1,10,90,360,1050,2016,2730,2520,1530,550,89

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

%C 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).

%C 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

%C 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

%C 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

%H G. C. Greubel, <a href="/A094436/b094436.txt">Rows n = 0..100 of triangle, flattened</a>

%H Michael Griffin, Ken Ono, Larry Rolen, and Don Zagier, <a href="https://doi.org/10.1073/pnas.1902572116">Jensen polynomials for the Riemann zeta function and other sequences</a>, PNAS, vol. 116, no. 23, 11103-11110, June 4, 2019.

%F 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

%F G.f. (-1+x)/(-1+2*x+x*y-x^2*y+x^2*y^2-x^2). - _R. J. Mathar_, Aug 11 2015

%F From _G. C. Greubel_, Oct 30 2019: (Start)

%F T(n, k) = binomial(n, k)*Fibonacci(k+1).

%F Sum_{k=0..n} T(n,k) = Fibonacci(2*n+1).

%F Sum_{k=0..n} (-1)^k*T(n,k) = Fibonacci(n-1). (End)

%e First four rows:

%e 1

%e 1 1

%e 1 2 2

%e 1 3 6 3

%e Sum = 1+3+6+3=13=F(7); alt.Sum = 1-3+6-3=1=F(2).

%e T(3,2)=F(3)C(3,2)=2*3=6.

%e From _Philippe Deléham_, Mar 26 2012: (Start)

%e (1, 0, 0, 1, 0, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, 0, ...) begins :

%e 1

%e 1, 0

%e 1, 1, 0

%e 1, 2, 2, 0

%e 1, 3, 6, 3, 0

%e 1, 4, 12, 12, 5, 0

%e 1, 5, 20, 30, 25, 8, 0

%e 1, 6, 30, 60, 75, 48, 13, 0 . (End)

%p with(combinat); seq(seq(fibonacci(k+1)*binomial(n,k), k=0..n), n=0..12); # _G. C. Greubel_, Oct 30 2019

%t (* First program *)

%t u[1, x_] := 1; v[1, x_] := 1; z = 13;

%t u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];

%t v[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x];

%t Table[Expand[u[n, x]], {n, 1, z/2}]

%t Table[Expand[v[n, x]], {n, 1, z/2}]

%t cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];

%t TableForm[cu]

%t Flatten[%] (* A094436 *)

%t Table[Expand[v[n, x]], {n, 1, z}]

%t cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];

%t TableForm[cv]

%t Flatten[%] (* A094437 *)

%t (* Second program *)

%t Table[Fibonacci[k+1]*Binomial[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jul 11 2019 *)

%o (PARI) T(n,k) = fibonacci(k+1)*binomial(n,k); \\ _G. C. Greubel_, Jul 11 2019

%o (Magma) [Fibonacci(k+1)*Binomial(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jul 11 2019

%o (Sage) [[fibonacci(k+1)*binomial(n,k) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Jul 11 2019

%o (GAP) Flat(List([0..12], n-> List([0..n], k-> Fibonacci(k+1)* Binomial(n,k) ))); # _G. C. Greubel_, Jul 11 2019

%Y Cf. A000045.

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

%K nonn,easy,tabl

%O 0,5

%A _Clark Kimberling_, May 03 2004

%E Offset set to 0 by _Alois P. Heinz_, Aug 11 2015