Real-rooted atlas

A360282 → basis A008277

nonnegative coefficients

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

Recurrence

\(\left(-3\,n + 2\,n^{2}\right)\,P_{n}(x) = \left(-2 - 6\,x - 2\,x^{2} + 5\,n + 15\,n x + 5\,n x^{2} - 2\,n^{2} - 6\,n^{2} x - 2\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(-6\,x - 4\,x^{2} + 15\,n x + 10\,n x^{2} - 6\,n^{2} x - 4\,n^{2} x^{2}\right)\,P_{n-2}'(x) + \left(-2\,x^{2} + 5\,n x^{2} - 2\,n^{2} x^{2}\right)\,P_{n-2}''(x) + \left(10 + 4\,x - 24\,n - 8\,n x + 12\,n^{2} + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(4\,x - 8\,n x + 4\,n^{2} x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1112
259216
32567303125
41294853407041028
568134913490113513558937
636532509534076157852975231681821
7198251803913228542015375397066503647785598

All 15 rows

Technical details
Node id
n-ea7b734e6d6f0c71 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-201b247ab711178c

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:      1,       1;
  2:      5,       9,        2;
  3:     25,      67,       30,        3;
  4:    129,     485,      340,       70,        4;
  5:    681,    3491,     3490,     1135,      135,       5;
  6:   3653,   25095,    34076,    15785,     2975,     231,      6;
  7:  19825,  180391,   322854,   201537,    53970,    6650,    364,     7;
  8: 108545, 1297413,  2998892,  2436798,   873978,  152670,  13272,   540,   8;
  9: 598417, 9339259, 27475950, 28382475, 13136403, 3059133, 376320, 24330, 765, 9;
%F Formula
The row polynomials satisfy (2*n^2-3*n)*P(n,x) = -(2*n^2-5*n+2+(6*n^2-15*n+6)*x+(2*n^2-5*n+2)*x^2)*P(n-2,x) - ((6*n^2-15*n+6)*x+(4*n^2-10*n+4)*x^2)*d/dx P(n-2,x) - (2*n^2-5*n+2)*x^2*d^2/dx^2 P(n-2,x) + (12*n^2-24*n+10+(4*n^2-8*n+4)*x)*P(n-1,x) + (4*n^2-8*n+4)*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^2 - 5*n + 2 + (6*n^2 - 15*n + 6)*x + (2*n^2 - 5*n + 2)*x^2)*P[n - 2, x] - ((6*n^2 - 15*n + 6)*x + (4*n^2 - 10*n + 4)*x^2)*D[P[n - 2, x], x] - (2*n^2 - 5*n + 2)*x^2*D[P[n - 2, x], {x, 2}] + (12*n^2 - 24*n + 10 + (4*n^2 - 8*n + 4)*x)*P[n - 1, x] + (4*n^2 - 8*n + 4)*x*D[P[n - 1, x], x])/(2*n^2 - 3*n)]];
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] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n^2 - 5*n + 2 + (6*n^2 - 15*n + 6)*x + (2*n^2 - 5*n + 2)*x^2)*v[i - 2] - ((6*n^2 - 15*n + 6)*x + (4*n^2 - 10*n + 4)*x^2)*deriv(v[i - 2], x) - (2*n^2 - 5*n + 2)*x^2*deriv(deriv(v[i - 2], x), x) + (12*n^2 - 24*n + 10 + (4*n^2 - 8*n + 4)*x)*v[i - 1] + (4*n^2 - 8*n + 4)*x*deriv(v[i - 1], x))/(2*n^2 - 3*n)); 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.