Real-rooted atlas

A111125 → reverse → θ+1

nonnegative coefficients

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

Recurrence

\(\left(6 - 6\,x - 10\,n + 22\,n x + 4\,n^{2} - 24\,n^{2} x + 8\,n^{3} x\right)\,P_{n}(x) = \left(6 + 30\,x + 24\,x^{2} - 10\,n - 38\,n x - 10\,n x^{2} + 4\,n^{2} - 40\,n^{2} x^{2} + 8\,n^{3} x + 24\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(2\,x + 11\,x^{2} + 12\,x^{3} - 2\,n x - 8\,n x^{2} - 4\,n^{2} x^{2} - 16\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2167
31101526
4114422885
51188112045265
612213230827566804
7126195624910546912393
81302701100225022689801207019

All 15 rows

Technical details
Node id
n-87557a136833d974 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-122247bb6f6b155e

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: 1,  6;
   3: 1, 10,  15;
   4: 1, 14,  42,   28;
   5: 1, 18,  81,  120,   45;
   6: 1, 22, 132,  308,  275,    66;
   7: 1, 26, 195,  624,  910,   546,    91;
   8: 1, 30, 270, 1100, 2250,  2268,   980,   120;
   9: 1, 34, 357, 1768, 4675,  6732,  4998,  1632,  153;
  10: 1, 38, 456, 2660, 8645, 16302, 17556, 10032, 2565, 190;
%F Formula
The row polynomials satisfy (4*n^2-10*n+6+(8*n^3-24*n^2+22*n-6)*x)*P(n,x) = (4*n^2-10*n+6+(8*n^3-38*n+30)*x+(24*n^3-40*n^2-10*n+24)*x^2)*P(n-1,x) - ((2*n-2)*x+(4*n^2+8*n-11)*x^2+(16*n^2-12)*x^3)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 - 10*n + 6 + (8*n^3 - 38*n + 30)*x + (24*n^3 - 40*n^2 - 10*n + 24)*x^2)*P[n - 1, x] - ((2*n - 2)*x + (4*n^2 + 8*n - 11)*x^2 + (16*n^2 - 12)*x^3)*D[P[n - 1, x], x])/(4*n^2 - 10*n + 6 + (8*n^3 - 24*n^2 + 22*n - 6)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((4*n^2 - 10*n + 6 + (8*n^3 - 38*n + 30)*x + (24*n^3 - 40*n^2 - 10*n + 24)*x^2)*v[i - 1] - ((2*n - 2)*x + (4*n^2 + 8*n - 11)*x^2 + (16*n^2 - 12)*x^3)*deriv(v[i - 1], x))/(4*n^2 - 10*n + 6 + (8*n^3 - 24*n^2 + 22*n - 6)*x)); 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

Same fingerprint ?

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023