Real-rooted atlas

A094638 → θ+1 → basis A008277

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1325
21324643
37027917724550
44463535430514001209806
53276499651047056140012170720232236
6272527896212690765253610087248011642450407037682
725329613891689744688071060686905635836512724908122043640320265026511

All 15 rows

Technical details
Node id
n-2efa4d45d0558754 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-583e3cf97543ca81

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:        3,          2;
  2:       13,         24,           6;
  3:       70,        279,         177,           24;
  4:      446,       3535,        4305,         1400,          120;
  5:     3276,      49965,      104705,        61400,        12170,          720;
  6:    27252,     789621,     2690765,      2536100,       872480,       116424,        5040;
  7:   253296,   13891689,    74468807,    106068690,     56358365,     12724908,     1220436,      40320;
  8:  2602224,  270294921,  2232714407,   4627581826,   3564813805,   1211143164,   193024244,   13940352,    362880;
  9: 29288160, 5776027245, 72546228995, 213388039480, 229120729999, 109885000974, 25873597568, 3063592224, 172535184, 3628800;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(n^3+(3*n^3-4*n^2+n)*x+(n^3-2*n^2+n)*x^2)*P(n-2,x) - ((3*n^3-4*n^2+n)*x+(2*n^3-4*n^2+2*n)*x^2)*d/dx P(n-2,x) - (n^3-2*n^2+n)*x^2*d^2/dx^2 P(n-2,x) + (2*n^2-1+(2*n^2-2*n)*x)*P(n-1,x) + (2*n^2-2*n)*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + (3*n^3 - 4*n^2 + n)*x + (n^3 - 2*n^2 + n)*x^2)*P[n - 2, x] - ((3*n^3 - 4*n^2 + n)*x + (2*n^3 - 4*n^2 + 2*n)*x^2)*D[P[n - 2, x], x] - (n^3 - 2*n^2 + n)*x^2*D[P[n - 2, x], {x, 2}] + (2*n^2 - 1 + (2*n^2 - 2*n)*x)*P[n - 1, x] + (2*n^2 - 2*n)*x*D[P[n - 1, x], x])/(n - 1)]];
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] = 3 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 + (3*n^3 - 4*n^2 + n)*x + (n^3 - 2*n^2 + n)*x^2)*v[i - 2] - ((3*n^3 - 4*n^2 + n)*x + (2*n^3 - 4*n^2 + 2*n)*x^2)*deriv(v[i - 2], x) - (n^3 - 2*n^2 + n)*x^2*deriv(deriv(v[i - 2], x), x) + (2*n^2 - 1 + (2*n^2 - 2*n)*x)*v[i - 1] + (2*n^2 - 2*n)*x*deriv(v[i - 1], x))/(n - 1)); 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. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  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. [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
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.