Real-rooted atlas

A281260 → basis A008277

nonnegative coefficients

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

Recurrence

\(\left(-2 + 3\,n + 2\,n^{2}\right)\,P_{n}(x) = \left(2\,x + 2\,x^{2} + 3\,n x + 3\,n x^{2} - 2\,n^{2} x - 2\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(2\,x + 4\,x^{2} + 3\,n x + 6\,n x^{2} - 2\,n^{2} x - 4\,n^{2} x^{2}\right)\,P_{n-2}'(x) + \left(2\,x^{2} + 3\,n x^{2} - 2\,n^{2} x^{2}\right)\,P_{n-2}''(x) + \left(-2 + 2\,x + 8\,n^{2} + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(2\,x + 4\,n^{2} x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1224
2511319
31454324104
442262260705639
51321280192485013064322
642963231365788902205217731728
714303159894968859883094049003368250168

All 15 rows

Technical details
Node id
n-6d84ed1513a96b5c · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-0cfc51b05cf62085

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 0)
  • 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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:     1;
  1:     2,      2;
  2:     5,     11,       3;
  3:    14,     54,      32,       4;
  4:    42,    262,     260,      70,       5;
  5:   132,   1280,    1924,     850,     130,       6;
  6:   429,   6323,   13657,    8890,    2205,     217,      7;
  7:  1430,  31598,   94968,   85988,   30940,    4900,    336,     8;
  8:  4862, 159650,  653592,  794262,  390369,   88788,   9744,   492,   9;
  9: 16796, 814824, 4476404, 7126570, 4607610, 1414854, 221424, 17820, 690, 10;
%F Formula
The row polynomials satisfy (2*n^2+3*n-2)*P(n,x) = -((2*n^2-3*n-2)*x+(2*n^2-3*n-2)*x^2)*P(n-2,x) - ((2*n^2-3*n-2)*x+(4*n^2-6*n-4)*x^2)*d/dx P(n-2,x) - (2*n^2-3*n-2)*x^2*d^2/dx^2 P(n-2,x) + (8*n^2-2+(4*n^2+2)*x)*P(n-1,x) + (4*n^2+2)*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-((2*n^2 - 3*n - 2)*x + (2*n^2 - 3*n - 2)*x^2)*P[n - 2, x] - ((2*n^2 - 3*n - 2)*x + (4*n^2 - 6*n - 4)*x^2)*D[P[n - 2, x], x] - (2*n^2 - 3*n - 2)*x^2*D[P[n - 2, x], {x, 2}] + (8*n^2 - 2 + (4*n^2 + 2)*x)*P[n - 1, x] + (4*n^2 + 2)*x*D[P[n - 1, x], x])/(2*n^2 + 3*n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-((2*n^2 - 3*n - 2)*x + (2*n^2 - 3*n - 2)*x^2)*v[i - 2] - ((2*n^2 - 3*n - 2)*x + (4*n^2 - 6*n - 4)*x^2)*deriv(v[i - 2], x) - (2*n^2 - 3*n - 2)*x^2*deriv(deriv(v[i - 2], x), x) + (8*n^2 - 2 + (4*n^2 + 2)*x)*v[i - 1] + (4*n^2 + 2)*x*deriv(v[i - 1], x))/(2*n^2 + 3*n - 2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

  • Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
    by Ray basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
    derived from Ray basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
    derived from Ray basis map theorem

Proof chains ? JSON

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. [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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.