Real-rooted atlas

A121574 → x ↦ x+1 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-26\,n + 21\,n x + 36\,n x^{2} + 5\,n x^{3} - 13\,n^{2} x + 17\,n^{2} x^{2} + 8\,n^{2} x^{3} - 3\,n^{3} x^{2} + 3\,n^{3} x^{3}\right)\,P_{n}(x) = \left(130 + 324\,x + 268\,x^{2} + 84\,x^{3} + 10\,x^{4} - 208\,n - 209\,n x + 44\,n x^{2} + 81\,n x^{3} + 16\,n x^{4} - 143\,n^{2} x - 122\,n^{2} x^{2} - 5\,n^{2} x^{3} + 6\,n^{2} x^{4} - 24\,n^{3} x^{2} - 12\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(169 + 533\,x + 622\,x^{2} + 326\,x^{3} + 73\,x^{4} + 5\,x^{5} + 169\,n x + 403\,n x^{2} + 327\,n x^{3} + 101\,n x^{4} + 8\,n x^{5} + 39\,n^{2} x^{2} + 69\,n^{2} x^{3} + 33\,n^{2} x^{4} + 3\,n^{2} x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1325
21014327
33370334140
4109310228605712
5360128412875329563564
611895100648036761030138717620
7392719682303032191684451770189886240

All 15 rows

Technical details
Node id
n-4320661fe2cc95aa · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-c3630c75eca728bc

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 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;
  2:    10,     14,      3;
  3:    33,     70,     33,      4;
  4:   109,    310,    228,     60,      5;
  5:   360,   1284,   1287,    532,     95,      6;
  6:  1189,   5100,   6480,   3676,   1030,    138,     7;
  7:  3927,  19682,  30303,  21916,   8445,   1770,   189,    8;
  8: 12970,  74378, 134562, 118468,  58355,  16818,  2800,  248,   9;
  9: 42837, 276610, 575079, 597140, 358990, 132384, 30275, 4168, 315, 10;
%F Formula
The row polynomials satisfy -(26*n+(13*n^2-21*n)*x+(3*n^3-17*n^2-36*n)*x^2-(3*n^3+8*n^2+5*n)*x^3)*P(n,x) = -(208*n-130+(143*n^2+209*n-324)*x+(24*n^3+122*n^2-44*n-268)*x^2+(12*n^3+5*n^2-81*n-84)*x^3-(6*n^2+16*n+10)*x^4)*P(n-1,x) + (169+(169*n+533)*x+(39*n^2+403*n+622)*x^2+(69*n^2+327*n+326)*x^3+(33*n^2+101*n+73)*x^4+(3*n^2+8*n+5)*x^5)*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[(-(208*n - 130 + (143*n^2 + 209*n - 324)*x + (24*n^3 + 122*n^2 - 44*n - 268)*x^2 + (12*n^3 + 5*n^2 - 81*n - 84)*x^3 - (6*n^2 + 16*n + 10)*x^4)*P[n - 1, x] + (169 + (169*n + 533)*x + (39*n^2 + 403*n + 622)*x^2 + (69*n^2 + 327*n + 326)*x^3 + (33*n^2 + 101*n + 73)*x^4 + (3*n^2 + 8*n + 5)*x^5)*D[P[n - 1, x], x])/(-26*n - (13*n^2 - 21*n)*x - (3*n^3 - 17*n^2 - 36*n)*x^2 + (3*n^3 + 8*n^2 + 5*n)*x^3)]];
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] = (-(208*n - 130 + (143*n^2 + 209*n - 324)*x + (24*n^3 + 122*n^2 - 44*n - 268)*x^2 + (12*n^3 + 5*n^2 - 81*n - 84)*x^3 - (6*n^2 + 16*n + 10)*x^4)*v[i - 1] + (169 + (169*n + 533)*x + (39*n^2 + 403*n + 622)*x^2 + (69*n^2 + 327*n + 326)*x^3 + (33*n^2 + 101*n + 73)*x^4 + (3*n^2 + 8*n + 5)*x^5)*deriv(v[i - 1], x))/(-26*n - (13*n^2 - 21*n)*x - (3*n^3 - 17*n^2 - 36*n)*x^2 + (3*n^3 + 8*n^2 + 5*n)*x^3)); 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.