Real-rooted atlas

A094816 → basis A013609 → D+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(-12 + 7\,n - n^{2}\right)\,P_{n-4}(x) + \left(9 - 12\,x - 9\,n + 4\,n x + 2\,n^{2}\right)\,P_{n-3}(x) + \left(4 + 4\,x - 4\,x^{2} - 4\,n x - n^{2}\right)\,P_{n-2}(x) + \left(-1 + 4\,x + 2\,n\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 4 + 2\,x\)
\(P_{3}(x) = 15 + 18\,x + 4\,x^{2}\)
\(P_{4}(x) = 62 + 118\,x + 60\,x^{2} + 8\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2426
31518437
462118608248
5297752596176161837
616645098536024404803215074
7108153793048496292408800124864136593
8804503121984607403389121349602912031361281359644

All 15 rows

Technical details
Node id
n-a18c6712c2ed5c72 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-5a4f854e1a92e4f8

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept -1)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:       1;
   2:       4,        2;
   3:      15,       18,        4;
   4:      62,      118,       60,        8;
   5:     297,      752,      596,      176,       16;
   6:    1664,     5098,     5360,     2440,      480,      32;
   7:   10815,    37930,    48496,    29240,     8800,    1248,      64;
   8:   80450,   312198,   460740,   338912,   134960,   29120,    3136,    128;
   9:  675745,  2838672,  4677032,  3992352,  1943984,  554624,   90496,   7680,   256;
  10: 6333052, 28372914, 51034272, 48926800, 27750912, 9697632, 2091264, 268032, 18432, 512;
%F Formula
The row polynomials satisfy P(n,x) = -(n^2-7*n+12)*P(n-4,x) + (2*n^2-9*n+9+(4*n-12)*x)*P(n-3,x) - (n^2-4+(4*n-4)*x+4*x^2)*P(n-2,x) + (2*n-1+4*x)*P(n-1,x) for n >= 5, with P(1,x) = 1, P(2,x) = 4 + 2*x, P(3,x) = 15 + 18*x + 4*x^2, P(4,x) = 62 + 118*x + 60*x^2 + 8*x^3.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 4 + 2*x;
P[3, x_] := 15 + 18*x + 4*x^2;
P[4, x_] := 62 + 118*x + 60*x^2 + 8*x^3;
P[n_, x_] := P[n, x] = Expand[-(n^2 - 7*n + 12)*P[n - 4, x] + (2*n^2 - 9*n + 9 + (4*n - 12)*x)*P[n - 3, x] - (n^2 - 4 + (4*n - 4)*x + 4*x^2)*P[n - 2, x] + (2*n - 1 + 4*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 4 + 2*x; v[3] = 15 + 18*x + 4*x^2; v[4] = 62 + 118*x + 60*x^2 + 8*x^3; for(n = 5, N, my(i = n); v[i] = -(n^2 - 7*n + 12)*v[i - 4] + (2*n^2 - 9*n + 9 + (4*n - 12)*x)*v[i - 3] - (n^2 - 4 + (4*n - 4)*x + 4*x^2)*v[i - 2] + (2*n - 1 + 4*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

References

  1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023