Real-rooted atlas

A181371 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-30\,n + 9\,n x + 15\,n^{2} x + 3\,n^{2} x^{2} + 3\,n^{3} x^{2}\right)\,P_{n}(x) = \left(-20 + 36\,x - 12\,x^{2} - 4\,x^{3} - 70\,n + 6\,n x - 6\,n x^{2} - 2\,n x^{3} + 30\,n^{2} x + 29\,n^{2} x^{2} + 4\,n^{2} x^{3} + 7\,n^{3} x^{2} + 2\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-25 + 20\,x + 27\,x^{2} - 14\,x^{3} - 8\,x^{4} + 25\,n x + 10\,n x^{2} - 23\,n x^{3} - 12\,n x^{4} + 5\,n^{2} x^{2} - n^{2} x^{3} - 4\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
133
28210
3211233
455503108
514418027351
637760015341134
79871908702483645
8258458782853344511664
96765177001071919207537179
1017711523903812192206506118098
1146368153000130194400324275108373977
1212139344203243104616170423700109871180980
133178111265832139244461867211686583161473720087
14832040359825044093342269040529025522901715811691702
Technical details
Node id
n-4c9cfea0cf9e83ee · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-dd535a5eb6454c1b

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half affine: slope 1, intercept 0)
  • 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:    1;
  1:    3;
  2:    8,     2;
  3:   21,    12;
  4:   55,    50,     3;
  5:  144,   180,    27;
  6:  377,   600,   153,    4;
  7:  987,  1908,   702,   48;
  8: 2584,  5878,  2853,  344,  5;
  9: 6765, 17700, 10719, 1920, 75;
%F Formula
The row polynomials satisfy -(30*n-(15*n^2+9*n)*x-(3*n^3+3*n^2)*x^2)*P(n,x) = -(70*n+20-(30*n^2+6*n+36)*x-(7*n^3+29*n^2-6*n-12)*x^2-(2*n^3+4*n^2-2*n-4)*x^3)*P(n-1,x) - (25-(25*n+20)*x-(5*n^2+10*n+27)*x^2+(n^2+23*n+14)*x^3+(4*n^2+12*n+8)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(70*n + 20 - (30*n^2 + 6*n + 36)*x - (7*n^3 + 29*n^2 - 6*n - 12)*x^2 - (2*n^3 + 4*n^2 - 2*n - 4)*x^3)*P[n - 1, x] - (25 - (25*n + 20)*x - (5*n^2 + 10*n + 27)*x^2 + (n^2 + 23*n + 14)*x^3 + (4*n^2 + 12*n + 8)*x^4)*D[P[n - 1, x], x])/(-30*n + (15*n^2 + 9*n)*x + (3*n^3 + 3*n^2)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(70*n + 20 - (30*n^2 + 6*n + 36)*x - (7*n^3 + 29*n^2 - 6*n - 12)*x^2 - (2*n^3 + 4*n^2 - 2*n - 4)*x^3)*v[i - 1] - (25 - (25*n + 20)*x - (5*n^2 + 10*n + 27)*x^2 + (n^2 + 23*n + 14)*x^3 + (4*n^2 + 12*n + 8)*x^4)*deriv(v[i - 1], x))/(-30*n + (15*n^2 + 9*n)*x + (3*n^3 + 3*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 - 1, v[i])))};
T(9)

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.