A322944 Coefficients of a family of orthogonal polynomials. Triangle read by rows, T(n, k) for 0 <= k <= n.

1, 1, 1, 2, 6, 1, 6, 38, 15, 1, 24, 272, 188, 28, 1, 120, 2200, 2340, 580, 45, 1, 720, 19920, 30280, 11040, 1390, 66, 1, 5040, 199920, 413560, 206920, 37450, 2842, 91, 1, 40320, 2204160, 5989760, 3931200, 955920, 102816, 5208, 120, 1

Offset

0,4

Comments

The polynomials represent a family of orthogonal polynomials which obey a recurrence of the form p(n, x) = (x+r(n))*p(n-1, x) - s(n)*p(n-2, x) + t(n)*p(n-3, x) - u(n)*p(n-4, x). For the details see the Maple program.

We conjecture that the polynomials have only negative and simple real roots.

Links

Table of n, a(n) for n=0..44.

Peter Luschny, Plot of the polynomials.

Formula

Let R be the inverse of the Riordan square [see A321620] of (1 - 3*x)^(-1/3) then T(n, k) = (-1)^(n-k)*R(n, k).

Example

Triangle starts:
[0]    1;
[1]    1,      1;
[2]    2,      6,      1;
[3]    6,     38,     15,      1;
[4]   24,    272,    188,     28,     1;
[5]  120,   2200,   2340,    580,    45,    1;
[6]  720,  19920,  30280,  11040,  1390,   66,  1;
[7] 5040, 199920, 413560, 206920, 37450, 2842, 91, 1;
Production matrix starts:
   1;
   1,    1;
   3,    5,    1;
   6,   18,    9,    1;
   6,   42,   45,   13,    1;
   0,   48,  132,   84,   17,    1;
   0,    0,  180,  300,  135,   21,    1;
   0,    0,    0,  480,  570,  198,   25,    1;

Maple

P := proc(n) option remember; local a, b, c, d;
a := n -> 4*n-3; b := n -> 3*(n-1)*(2*n-3);
c := n -> (n-1)*(n-2)*(4*n-9); d := n -> (n-2)*(n-1)*(n-3)^2;
if n = 0 then return 1 fi;
if n = 1 then return x + 1 fi;
if n = 2 then return x^2 + 6*x + 2 fi;
if n = 3 then return x^3 + 15*x^2 + 38*x + 6 fi;
expand((x+a(n))*P(n-1) - b(n)*P(n-2) + c(n)*P(n-3) - d(n)*P(n-4)) end:
seq(print(P(n)), n=0..9); # Computes the polynomials.

Mathematica

a[n_] := 4n - 3;
b[n_] := 3(n - 1)(2n - 3);
c[n_] := (n - 1)(n - 2)(4n - 9);
d[n_] := (n - 2)(n - 1)(n - 3)^2;
P[n_] := P[n] = Switch[n, 0, 1, 1, x + 1, 2, x^2 + 6x + 2, 3, x^3 + 15x^2 + 38x + 6, _, Expand[(x + a[n]) P[n - 1] - b[n] P[n - 2] + c[n] P[n - 3] - d[n] P[n - 4]]];
Table[CoefficientList[P[n], x], {n, 0, 9}] (* Jean-François Alcover, Jun 15 2019, from Maple *)

Prog

(Sage) # uses[riordan_square from A321620]
R = riordan_square((1 - 3*x)^(-1/3), 9, True).inverse()
for n in (0..8): print([(-1)^(n-k)*c for (k, c) in enumerate(R.row(n)[:n+1])])

Crossrefs

p(n, 1) = A322943(n) (row sums); p(n, 0) = n! = A000142(n).

A321966 (m=2), this sequence (m=3).

Cf. A321620.

Sequence in context: A329207 A191100 A364708 * A019576 A141906 A352125

Adjacent sequences: A322941 A322942 A322943 * A322945 A322946 A322947

Keyword

nonn,tabl

Author

Peter Luschny, Jan 02 2019

Status

approved