Real-rooted atlas

A209599 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(40 + 38\,x + 12\,x^{2} - 10\,n + 33\,n x + 20\,n x^{2} + 2\,n x^{3} - 5\,n^{2} x + 11\,n^{2} x^{2} + 3\,n^{2} x^{3} - n^{3} x^{2} + n^{3} x^{3}\right)\,P_{n}(x) = \left(90 + 78\,x - 36\,x^{2} - 38\,x^{3} - 4\,x^{4} - 30\,n + 74\,n x + 108\,n x^{2} + 21\,n x^{3} - 2\,n x^{4} - 20\,n^{2} x + 15\,n^{2} x^{2} + 28\,n^{2} x^{3} + 4\,n^{2} x^{4} - 3\,n^{3} x^{2} + n^{3} x^{3} + 2\,n^{3} x^{4}\right)\,P_{n-1}(x) + \left(25 + 5\,x - 102\,x^{2} - 132\,x^{3} - 58\,x^{4} - 8\,x^{5} + 25\,n x + 20\,n x^{2} - 40\,n x^{3} - 47\,n x^{4} - 12\,n x^{5} + 5\,n^{2} x^{2} + 4\,n^{2} x^{3} - 5\,n^{2} x^{4} - 4\,n^{2} x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
011
122
2325
35611
4814325
513301255
62160364121
7341169320263

All 15 rows

Technical details
Node id
n-4a1071f2dabe7b56 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-8495a6d14838bc51

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:  2;
  2:  3,   2;
  3:  5,   6;
  4:  8,  14,   3;
  5: 13,  30,  12;
  6: 21,  60,  36,   4;
  7: 34, 116,  93,  20;
  8: 55, 218, 219,  72,  5;
  9: 89, 402, 486, 216, 30;
%F Formula
The row polynomials satisfy -(10*n-40+(5*n^2-33*n-38)*x+(n^3-11*n^2-20*n-12)*x^2-(n^3+3*n^2+2*n)*x^3)*P(n,x) = -(30*n-90+(20*n^2-74*n-78)*x+(3*n^3-15*n^2-108*n+36)*x^2-(n^3+28*n^2+21*n-38)*x^3-(2*n^3+4*n^2-2*n-4)*x^4)*P(n-1,x) + (25+(25*n+5)*x+(5*n^2+20*n-102)*x^2+(4*n^2-40*n-132)*x^3-(5*n^2+47*n+58)*x^4-(4*n^2+12*n+8)*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[(-(30*n - 90 + (20*n^2 - 74*n - 78)*x + (3*n^3 - 15*n^2 - 108*n + 36)*x^2 - (n^3 + 28*n^2 + 21*n - 38)*x^3 - (2*n^3 + 4*n^2 - 2*n - 4)*x^4)*P[n - 1, x] + (25 + (25*n + 5)*x + (5*n^2 + 20*n - 102)*x^2 + (4*n^2 - 40*n - 132)*x^3 - (5*n^2 + 47*n + 58)*x^4 - (4*n^2 + 12*n + 8)*x^5)*D[P[n - 1, x], x])/(-10*n + 40 - (5*n^2 - 33*n - 38)*x - (n^3 - 11*n^2 - 20*n - 12)*x^2 + (n^3 + 3*n^2 + 2*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] = (-(30*n - 90 + (20*n^2 - 74*n - 78)*x + (3*n^3 - 15*n^2 - 108*n + 36)*x^2 - (n^3 + 28*n^2 + 21*n - 38)*x^3 - (2*n^3 + 4*n^2 - 2*n - 4)*x^4)*v[i - 1] + (25 + (25*n + 5)*x + (5*n^2 + 20*n - 102)*x^2 + (4*n^2 - 40*n - 132)*x^3 - (5*n^2 + 47*n + 58)*x^4 - (4*n^2 + 12*n + 8)*x^5)*deriv(v[i - 1], x))/(-10*n + 40 - (5*n^2 - 33*n - 38)*x - (n^3 - 11*n^2 - 20*n - 12)*x^2 + (n^3 + 3*n^2 + 2*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.