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

%I A322944 #20 Mar 25 2020 07:55:33
%S A322944 1,1,1,2,6,1,6,38,15,1,24,272,188,28,1,120,2200,2340,580,45,1,720,
%T A322944 19920,30280,11040,1390,66,1,5040,199920,413560,206920,37450,2842,91,
%U A322944 1,40320,2204160,5989760,3931200,955920,102816,5208,120,1
%N A322944 Coefficients of a family of orthogonal polynomials. Triangle read by rows, T(n, k) for 0 <= k <= n.
%C A322944 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.
%C A322944 We conjecture that the polynomials have only negative and simple real roots.
%H A322944 Peter Luschny, <a href="/A322944/a322944.jpg">Plot of the polynomials</a>.
%F A322944 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).
%e A322944 Triangle starts:
%e A322944 [0]    1;
%e A322944 [1]    1,      1;
%e A322944 [2]    2,      6,      1;
%e A322944 [3]    6,     38,     15,      1;
%e A322944 [4]   24,    272,    188,     28,     1;
%e A322944 [5]  120,   2200,   2340,    580,    45,    1;
%e A322944 [6]  720,  19920,  30280,  11040,  1390,   66,  1;
%e A322944 [7] 5040, 199920, 413560, 206920, 37450, 2842, 91, 1;
%e A322944 Production matrix starts:
%e A322944    1;
%e A322944    1,    1;
%e A322944    3,    5,    1;
%e A322944    6,   18,    9,    1;
%e A322944    6,   42,   45,   13,    1;
%e A322944    0,   48,  132,   84,   17,    1;
%e A322944    0,    0,  180,  300,  135,   21,    1;
%e A322944    0,    0,    0,  480,  570,  198,   25,    1;
%p A322944 P := proc(n) option remember; local a, b, c, d;
%p A322944 a := n -> 4*n-3; b := n -> 3*(n-1)*(2*n-3);
%p A322944 c := n -> (n-1)*(n-2)*(4*n-9); d := n -> (n-2)*(n-1)*(n-3)^2;
%p A322944 if n = 0 then return 1 fi;
%p A322944 if n = 1 then return x + 1 fi;
%p A322944 if n = 2 then return x^2 + 6*x + 2 fi;
%p A322944 if n = 3 then return x^3 + 15*x^2 + 38*x + 6 fi;
%p A322944 expand((x+a(n))*P(n-1) - b(n)*P(n-2) + c(n)*P(n-3) - d(n)*P(n-4)) end:
%p A322944 seq(print(P(n)), n=0..9); # Computes the polynomials.
%t A322944 a[n_] := 4n - 3;
%t A322944 b[n_] := 3(n - 1)(2n - 3);
%t A322944 c[n_] := (n - 1)(n - 2)(4n - 9);
%t A322944 d[n_] := (n - 2)(n - 1)(n - 3)^2;
%t A322944 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]]];
%t A322944 Table[CoefficientList[P[n], x], {n, 0, 9}] (* _Jean-François Alcover_, Jun 15 2019, from Maple *)
%o A322944 (Sage) # uses[riordan_square from A321620]
%o A322944 R = riordan_square((1 - 3*x)^(-1/3), 9, True).inverse()
%o A322944 for n in (0..8): print([(-1)^(n-k)*c for (k, c) in enumerate(R.row(n)[:n+1])])
%Y A322944 p(n, 1) = A322943(n) (row sums); p(n, 0) = n! = A000142(n).
%Y A322944 A321966 (m=2), this sequence (m=3).
%Y A322944 Cf. A321620.
%K A322944 nonn,tabl
%O A322944 0,4
%A A322944 _Peter Luschny_, Jan 02 2019