Real-rooted atlas

A062145 → x ↦ x+1 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-2 + 4\,x - 3\,n - 4\,n x + 3\,n^{2} - 3\,n^{2} x + 2\,n^{3} + 2\,n^{3} x + n^{4} x\right)\,P_{n}(x) = \left(4\,x - 2\,x^{2} + 2\,n + 6\,n x + 9\,n x^{2} + 8\,n^{2} + 4\,n^{2} x + 8\,n^{3} + 18\,n^{3} x + 8\,n^{3} x^{2} + 4\,n^{4} x + 3\,n^{4} x^{2}\right)\,P_{n-1}(x) + \left(-x - 2\,x^{2} - x^{3} - 4\,n x - 4\,n x^{2} - 4\,n^{2} x - 10\,n^{2} x^{2} - 6\,n^{2} x^{3} - 2\,n^{3} x^{2} - 2\,n^{3} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
25813
3216030111
48433631580815
53301680226811201755573
6128779201386010080315033636633
75005360367722073920346507560588234979
81944816016040540548048030030099792161709601482715

All 15 rows

Technical details
Node id
n-6b0c5933048b26bb · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-c7092850c22e3de4

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:      5,       8;
   3:     21,      60,      30;
   4:     84,     336,     315,       80;
   5:    330,    1680,    2268,     1120,      175;
   6:   1287,    7920,   13860,    10080,     3150,     336;
   7:   5005,   36036,   77220,    73920,    34650,    7560,     588;
   8:  19448,  160160,  405405,   480480,   300300,   99792,   16170,    960;
   9:  75582,  700128, 2042040,  2882880,  2252250, 1009008,  252252,  31680,  1485;
  10: 293930, 3023280, 9976824, 16336320, 15315300, 8648640, 2942940, 576576, 57915, 2200;
%F Formula
The row polynomials satisfy (2*n^3+3*n^2-3*n-2+(n^4+2*n^3-3*n^2-4*n+4)*x)*P(n,x) = (8*n^3+8*n^2+2*n+(4*n^4+18*n^3+4*n^2+6*n+4)*x+(3*n^4+8*n^3+9*n-2)*x^2)*P(n-1,x) - ((4*n^2+4*n+1)*x+(2*n^3+10*n^2+4*n+2)*x^2+(2*n^3+6*n^2+1)*x^3)*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[((8*n^3 + 8*n^2 + 2*n + (4*n^4 + 18*n^3 + 4*n^2 + 6*n + 4)*x + (3*n^4 + 8*n^3 + 9*n - 2)*x^2)*P[n - 1, x] - ((4*n^2 + 4*n + 1)*x + (2*n^3 + 10*n^2 + 4*n + 2)*x^2 + (2*n^3 + 6*n^2 + 1)*x^3)*D[P[n - 1, x], x])/(2*n^3 + 3*n^2 - 3*n - 2 + (n^4 + 2*n^3 - 3*n^2 - 4*n + 4)*x)]];
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] = ((8*n^3 + 8*n^2 + 2*n + (4*n^4 + 18*n^3 + 4*n^2 + 6*n + 4)*x + (3*n^4 + 8*n^3 + 9*n - 2)*x^2)*v[i - 1] - ((4*n^2 + 4*n + 1)*x + (2*n^3 + 10*n^2 + 4*n + 2)*x^2 + (2*n^3 + 6*n^2 + 1)*x^3)*deriv(v[i - 1], x))/(2*n^3 + 3*n^2 - 3*n - 2 + (n^4 + 2*n^3 - 3*n^2 - 4*n + 4)*x)); 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.