OFFSET
0,5
COMMENTS
The name 'Jacobsthal triangle' used here is not standard.
LINKS
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
FORMULA
J(n, x) = (x+1)*J(n-1, x) + 2*J(n-2, x) for n >= 3.
T(n, k) = [x^k] J(n, x).
Equals the Riordan square (cf. A321620) generated by (2*x^2-1)/((x + 1)*(2*x - 1)).
Sum_{k=0..n} T(n, k) = A152035(n).
Sum_{k=0..n} (-1)^k*T(n, k) = A000007(n).
From G. C. Greubel, Sep 20 2023: (Start)
T(n, k) = [x^k]( [n=0] + (i*sqrt(2))^n*(ChebyshevU(n, (x+1)/(2*sqrt(2)*i)) + ChebyshevU(n-2, (x+1)/(2*sqrt(2)*i))) ).
G.f.: (1 - 2*t^2)/(1 - (x+1)*t - 2*t^2).
Sum_{k=0..floor(n/2)} T(n-k, k) = (2/3)*[n=0] + A006138(n-1).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = 2*[n=0] + Fibonacci(n-2). (End)
EXAMPLE
The first few polynomials are:
J(0, x) = 1;
J(1, x) = x + 1;
J(2, x) = x^2 + 2*x + 1;
J(3, x) = x^3 + 3*x^2 + 5*x + 3;
J(4, x) = x^4 + 4*x^3 + 10*x^2 + 12*x + 5;
J(5, x) = x^5 + 5*x^4 + 16*x^3 + 28*x^2 + 27*x + 11;
J(6, x) = x^6 + 6*x^5 + 23*x^4 + 52*x^3 + 75*x^2 + 62*x + 21;
The triangle starts:
[0] 1;
[1] 1, 1;
[2] 1, 2, 1;
[3] 3, 5, 3, 1;
[4] 5, 12, 10, 4, 1;
[5] 11, 27, 28, 16, 5, 1;
[6] 21, 62, 75, 52, 23, 6, 1;
[7] 43, 137, 193, 159, 85, 31, 7, 1;
[8] 85, 304, 480, 456, 290, 128, 40, 8, 1;
[9] 171, 663, 1170, 1254, 916, 480, 182, 50, 9, 1;
MAPLE
J := proc(n) option remember;
`if`(n < 3, [1, x+1, x^2 + 2*x + 1][n+1], (x+1)*J(n-1) + 2*J(n-2));
sort(expand(%)) end: seq(print(J(n)), n=0..11); # Computes the polynomials.
seq(seq(coeff(J(n), x, k), k=0..n), n=0..11);
MATHEMATICA
J[n_, x_]:= J[n, x]= If[n<3, (x+1)^n, (x+1)*J[n-1, x] + 2*J[n-2, x]];
T[n_, k_]:= Coefficient[J[n, x], x, k];
Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* Jean-François Alcover, Jun 17 2019 *)
(* Second program *)
A322942[n_, k_]:= Coefficient[Series[Boole[n==0] + (I*Sqrt[2])^n*(ChebyshevU[n, (x+1)/(2*Sqrt[2]*I)] + ChebyshevU[n-2, (x+ 1)/(2*Sqrt[2]*I)]), {x, 0, 50}], x, k];
Table[A322942[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Sep 20 2023 *)
PROG
(Sage) # use[riordan_square from A321620]
riordan_square((2*x^2 - 1)/((x + 1)*(2*x - 1)), 9)
(Magma)
R<x>:=PowerSeriesRing(Rationals(), 30);
g:= func< n, x | (&+[Binomial(n-k, k)*2^k*(x+1)^(n-2*k): k in [0..Floor(n/2)]]) >;
f:= func< n, x | n le 1 select (x+1)^n else g(n, x) - 2*g(n-2, x) >;
A322942:= func< n, k | Coefficient(R!( f(n, x) ), k) >;
[A322942(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 20 2023
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Jan 03 2019
STATUS
approved