Real-rooted atlas

A062145 → x ↦ x+1 → D+1

nonnegative coefficients

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

Recurrence

\(\left(6 + 2\,x - 2\,x^{2} - 5\,n - 5\,n x + n x^{2} - 4\,n^{2} + n^{2} x + n^{2} x^{2} + 2\,n^{3} + 2\,n^{3} x + n^{4}\right)\,P_{n}(x) = \left(-4\,x + 2\,x^{3} + 2\,n + 10\,n x + 6\,n x^{2} + n x^{3} - 10\,n^{2} - 5\,n^{2} x + 6\,n^{2} x^{2} + 3\,n^{2} x^{3} + 10\,n^{3} + 14\,n^{3} x + 6\,n^{3} x^{2} + 4\,n^{4} + 3\,n^{4} x\right)\,P_{n-1}(x) + \left(-x^{2} - 2\,x^{3} - x^{4} + 2\,n x - 2\,n x^{2} - 6\,n x^{3} - 2\,n x^{4} - 6\,n^{2} x - 10\,n^{2} x^{2} - 4\,n^{2} x^{3} - 2\,n^{3} x - 2\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
29413
3515010111
425237816520815
5117023521596420355573
65247132001218050409105636633
72302369498811804620013230176484234979
8995283503504954953603601432203049231501201482715

All 15 rows

Technical details
Node id
n-2b9d2aa1289243cb · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-9cc6669a2742dc83

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:       9,       4;
   3:      51,      50,       10;
   4:     252,     378,      165,       20;
   5:    1170,    2352,     1596,      420,       35;
   6:    5247,   13200,    12180,     5040,      910,      56;
   7:   23023,   69498,    81180,    46200,    13230,    1764,     84;
   8:   99528,  350350,   495495,   360360,   143220,   30492,   3150,    120;
   9:  425646, 1711424,  2842840,  2522520,  1291290,  384384,  63756,   5280,  165;
  10: 1805570, 8162856, 15577848, 16336320, 10270260, 3963960, 924924, 123552, 8415, 220;
%F Formula
The row polynomials satisfy (n^4+2*n^3-4*n^2-5*n+6+(2*n^3+n^2-5*n+2)*x+(n^2+n-2)*x^2)*P(n,x) = (4*n^4+10*n^3-10*n^2+2*n+(3*n^4+14*n^3-5*n^2+10*n-4)*x+(6*n^3+6*n^2+6*n)*x^2+(3*n^2+n+2)*x^3)*P(n-1,x) - ((2*n^3+6*n^2-2*n)*x+(2*n^3+10*n^2+2*n+1)*x^2+(4*n^2+6*n+2)*x^3+(2*n+1)*x^4)*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^4 + 10*n^3 - 10*n^2 + 2*n + (3*n^4 + 14*n^3 - 5*n^2 + 10*n - 4)*x + (6*n^3 + 6*n^2 + 6*n)*x^2 + (3*n^2 + n + 2)*x^3)*P[n - 1, x] - ((2*n^3 + 6*n^2 - 2*n)*x + (2*n^3 + 10*n^2 + 2*n + 1)*x^2 + (4*n^2 + 6*n + 2)*x^3 + (2*n + 1)*x^4)*D[P[n - 1, x], x])/(n^4 + 2*n^3 - 4*n^2 - 5*n + 6 + (2*n^3 + n^2 - 5*n + 2)*x + (n^2 + n - 2)*x^2)]];
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^4 + 10*n^3 - 10*n^2 + 2*n + (3*n^4 + 14*n^3 - 5*n^2 + 10*n - 4)*x + (6*n^3 + 6*n^2 + 6*n)*x^2 + (3*n^2 + n + 2)*x^3)*v[i - 1] - ((2*n^3 + 6*n^2 - 2*n)*x + (2*n^3 + 10*n^2 + 2*n + 1)*x^2 + (4*n^2 + 6*n + 2)*x^3 + (2*n + 1)*x^4)*deriv(v[i - 1], x))/(n^4 + 2*n^3 - 4*n^2 - 5*n + 6 + (2*n^3 + n^2 - 5*n + 2)*x + (n^2 + n - 2)*x^2)); 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

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.