Real-rooted atlas

A281000 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-2\,n - 4\,n^{2} - n^{2} x - n^{3} x + 2\,n^{4} x\right)\,P_{n}(x) = \left(4 - 8\,x - 2\,x^{2} + 34\,n x + 7\,n x^{2} - 16\,n^{2} - 26\,n^{2} x - 5\,n^{2} x^{2} - 8\,n^{3} x - 6\,n^{3} x^{2} + 8\,n^{4} x + 6\,n^{4} x^{2}\right)\,P_{n-1}(x) + \left(8\,x + 2\,x^{2} - x^{3} - 32\,n x - 4\,n x^{2} + 3\,n x^{3} + 4\,n^{2} x^{2} - n^{2} x^{3} + 16\,n^{3} x^{2} - 4\,n^{3} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2325
31020333
435140634242
512684075614451871
6462462069302640275614933
7171624024540543432071504687121739
864351201203783783603601251251638073581007541

All 15 rows

Technical details
Node id
n-ddf6aec16b6c760d · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-6aa82b38f20c7d91

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:     3,       2;
   3:    10,      20,        3;
   4:    35,     140,       63,        4;
   5:   126,     840,      756,      144,        5;
   6:   462,    4620,     6930,     2640,      275,       6;
   7:  1716,   24024,    54054,    34320,     7150,     468,      7;
   8:  6435,  120120,   378378,   360360,   125125,   16380,    735,     8;
   9: 24310,  583440,  2450448,  3267264,  1701700,  371280,  33320,  1088,    9;
  10: 92378, 2771340, 14965236, 26604864, 19399380, 6348888, 949620, 62016, 1539, 10;
%F Formula
The row polynomials satisfy -(4*n^2+2*n-(2*n^4-n^3-n^2)*x)*P(n,x) = -(16*n^2-4-(8*n^4-8*n^3-26*n^2+34*n-8)*x-(6*n^4-6*n^3-5*n^2+7*n-2)*x^2)*P(n-1,x) - ((32*n-8)*x-(16*n^3+4*n^2-4*n+2)*x^2+(4*n^3+n^2-3*n+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[(-(16*n^2 - 4 - (8*n^4 - 8*n^3 - 26*n^2 + 34*n - 8)*x - (6*n^4 - 6*n^3 - 5*n^2 + 7*n - 2)*x^2)*P[n - 1, x] - ((32*n - 8)*x - (16*n^3 + 4*n^2 - 4*n + 2)*x^2 + (4*n^3 + n^2 - 3*n + 1)*x^3)*D[P[n - 1, x], x])/(-4*n^2 - 2*n + (2*n^4 - n^3 - n^2)*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] = (-(16*n^2 - 4 - (8*n^4 - 8*n^3 - 26*n^2 + 34*n - 8)*x - (6*n^4 - 6*n^3 - 5*n^2 + 7*n - 2)*x^2)*v[i - 1] - ((32*n - 8)*x - (16*n^3 + 4*n^2 - 4*n + 2)*x^2 + (4*n^3 + n^2 - 3*n + 1)*x^3)*deriv(v[i - 1], x))/(-4*n^2 - 2*n + (2*n^4 - n^3 - n^2)*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.