Real-rooted atlas

A062145 → D → D+1

nonnegative coefficients

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

Recurrence

\(\left(-4 + 10\,x - 2\,x^{2} + 4\,n + n x - n x^{2} + 3\,n^{2} - 7\,n^{2} x + n^{2} x^{2} - 4\,n^{3} + 2\,n^{3} x + n^{4}\right)\,P_{n}(x) = \left(4\,x - 10\,x^{2} + 2\,x^{3} + 2\,n - 2\,n x + 17\,n x^{2} - 3\,n x^{3} - 3\,n^{2} + 5\,n^{2} x - 17\,n^{2} x^{2} + 3\,n^{2} x^{3} - 8\,n^{3} x + 6\,n^{3} x^{2} + n^{4} + 3\,n^{4} x\right)\,P_{n-1}(x) + \left(2\,x - 2\,x^{2} - n x - 5\,n x^{2} + 8\,n x^{3} - 2\,n x^{4} - 3\,n^{2} x + 7\,n^{2} x^{2} - 4\,n^{2} x^{3} + 2\,n^{3} x - 2\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 4\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
10
244
3302050
410821060378
528010928401402352
660039205880252028013200
71134111602772022680630050469498
81960270901008001352407056013860840350350

All 15 rows

Technical details
Node id
n-83b2e42137eca7d5 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-b39488b23a0bf731

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 -2)
  • 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:    0;
   2:    4;
   3:   30,     20;
   4:  108,    210,     60;
   5:  280,   1092,    840,     140;
   6:  600,   3920,   5880,    2520,     280;
   7: 1134,  11160,  27720,   22680,    6300,     504;
   8: 1960,  27090, 100800,  135240,   70560,   13860,    840;
   9: 3168,  58520, 304920,  609840,  517440,  188496,  27720,  1320;
  10: 4860, 115632, 803880, 2245320, 2827440, 1663200, 449064, 51480, 1980;
%F Formula
The row polynomials satisfy (n^4-4*n^3+3*n^2+4*n-4+(2*n^3-7*n^2+n+10)*x+(n^2-n-2)*x^2)*P(n,x) = (n^4-3*n^2+2*n+(3*n^4-8*n^3+5*n^2-2*n+4)*x+(6*n^3-17*n^2+17*n-10)*x^2+(3*n^2-3*n+2)*x^3)*P(n-1,x) + ((2*n^3-3*n^2-n+2)*x-(2*n^3-7*n^2+5*n+2)*x^2-(4*n^2-8*n)*x^3-2*n*x^4)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 4.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[((n^4 - 3*n^2 + 2*n + (3*n^4 - 8*n^3 + 5*n^2 - 2*n + 4)*x + (6*n^3 - 17*n^2 + 17*n - 10)*x^2 + (3*n^2 - 3*n + 2)*x^3)*P[n - 1, x] + ((2*n^3 - 3*n^2 - n + 2)*x - (2*n^3 - 7*n^2 + 5*n + 2)*x^2 - (4*n^2 - 8*n)*x^3 - 2*n*x^4)*D[P[n - 1, x], x])/(n^4 - 4*n^3 + 3*n^2 + 4*n - 4 + (2*n^3 - 7*n^2 + n + 10)*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] = 0; v[2] = 4; for(n = 3, N, my(i = n); v[i] = ((n^4 - 3*n^2 + 2*n + (3*n^4 - 8*n^3 + 5*n^2 - 2*n + 4)*x + (6*n^3 - 17*n^2 + 17*n - 10)*x^2 + (3*n^2 - 3*n + 2)*x^3)*v[i - 1] + ((2*n^3 - 3*n^2 - n + 2)*x - (2*n^3 - 7*n^2 + 5*n + 2)*x^2 - (4*n^2 - 8*n)*x^3 - 2*n*x^4)*deriv(v[i - 1], x))/(n^4 - 4*n^3 + 3*n^2 + 4*n - 4 + (2*n^3 - 7*n^2 + n + 10)*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.