Real-rooted atlas

A130850 → D

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
111
2325
31214329
460100454209
536078054012451809
62520672063002408315618269
72016063840756004082496607627210853
818144066528095256066729623310036300178582737769
9181440075600001270080010956960510300012277801306204088939497657
101995840093139200179625600184282560107692200355291206121500456016920710626813813
112395008001237420800269438400032059843202258625600953029440231739200293520481557468204701110851614157
12311351040017643225600428107680005798048256047870222400246906316807829902080144163219213675662052325004504512203522409089
1343589145600268799731200719220902400109263458304010352422080006323885568002486195712006089891808086562075606235047001735734098292134110690784225
1465383718400043589145600001274982508800021474919065600229751121600001623972430080076426301952002354350078080454348734840505840608002795492700570024242129791488957098135437
Technical details
Node id
n-64ff1752fc416ed2 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-ef5562416e43169f

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:       0;
  1:       1;
  2:       3,       2;
  3:      12,      14,        3;
  4:      60,     100,       45,        4;
  5:     360,     780,      540,      124,       5;
  6:    2520,    6720,     6300,     2408,     315,       6;
  7:   20160,   63840,    75600,    40824,    9660,     762,      7;
  8:  181440,  665280,   952560,   667296,  233100,   36300,   1785,    8;
  9: 1814400, 7560000, 12700800, 10956960, 5103000, 1227780, 130620, 4088, 9;
%F Formula
The row polynomials satisfy P(n,x) = -(n^2-2*n+(2*n^2-4*n)*x+(n^2-2*n)*x^2)*P(n-2,x) + (n*x+2*n*x^2+n*x^3)*d/dx P(n-2,x) + (2*n-1+(2*n-2)*x)*P(n-1,x) - (x+x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(n^2 - 2*n + (2*n^2 - 4*n)*x + (n^2 - 2*n)*x^2)*P[n - 2, x] + (n*x + 2*n*x^2 + n*x^3)*D[P[n - 2, x], x] + (2*n - 1 + (2*n - 2)*x)*P[n - 1, x] - (x + x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 - 2*n + (2*n^2 - 4*n)*x + (n^2 - 2*n)*x^2)*v[i - 2] + (n*x + 2*n*x^2 + n*x^3)*deriv(v[i - 2], x) + (2*n - 1 + (2*n - 2)*x)*v[i - 1] - (x + x^2)*deriv(v[i - 1], x)); 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

Proof chains ? JSON

Same fingerprint ?

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.