Real-rooted atlas

A099097 → reverse → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-27\,x - 243\,n + 189\,n x - 3\,n x^{2} + 243\,n^{2} - 243\,n^{2} x + 9\,n^{2} x^{2} + 81\,n^{3} x - 9\,n^{3} x^{2} + 3\,n^{4} x^{2}\right)\,P_{n}(x) = \left(-81\,x + 72\,x^{2} - 729\,n - 729\,n x - 9\,n x^{2} - 4\,n x^{3} + 729\,n^{2} - 81\,n^{2} x - 171\,n^{2} x^{2} - 6\,n^{2} x^{3} + 243\,n^{3} x + 63\,n^{3} x^{2} + 9\,n^{4} x^{2} + 2\,n^{4} x^{3}\right)\,P_{n-1}(x) + \left(81\,x^{2} + 36\,x^{3} - 729\,n x - 324\,n x^{2} - 9\,n x^{3} - 4\,n x^{4} - 243\,n^{2} x^{2} - 126\,n^{2} x^{3} - 8\,n^{2} x^{4} - 9\,n^{3} x^{3} - 4\,n^{3} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
233
39413
4272451
5811089198
624343281756
77291620486162851
821875832243019210641
96561204121093514402539373
101968369984459278640375144609
115904923619618370845360337536527724
12177147787320708588217728236256481915056
13531441259815626572059797761417506804496915181
14159432385030569743085419904076545054432102924860415
1547829692763493235075106173210403827250367416123486489021125
Technical details
Node id
n-34c3fca0a85f38f6 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-ee33b1416b29d17c

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 -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;
   3:     9,     4;
   4:    27,    24;
   5:    81,   108,     9;
   6:   243,   432,    81;
   7:   729,  1620,   486,   16;
   8:  2187,  5832,  2430,  192;
   9:  6561, 20412, 10935, 1440,  25;
  10: 19683, 69984, 45927, 8640, 375;
%F Formula
The row polynomials satisfy (243*n^2-243*n+(81*n^3-243*n^2+189*n-27)*x+(3*n^4-9*n^3+9*n^2-3*n)*x^2)*P(n,x) = (729*n^2-729*n+(243*n^3-81*n^2-729*n-81)*x+(9*n^4+63*n^3-171*n^2-9*n+72)*x^2+(2*n^4-6*n^2-4*n)*x^3)*P(n-1,x) - (729*n*x+(243*n^2+324*n-81)*x^2+(9*n^3+126*n^2+9*n-36)*x^3+(4*n^3+8*n^2+4*n)*x^4)*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[((729*n^2 - 729*n + (243*n^3 - 81*n^2 - 729*n - 81)*x + (9*n^4 + 63*n^3 - 171*n^2 - 9*n + 72)*x^2 + (2*n^4 - 6*n^2 - 4*n)*x^3)*P[n - 1, x] - (729*n*x + (243*n^2 + 324*n - 81)*x^2 + (9*n^3 + 126*n^2 + 9*n - 36)*x^3 + (4*n^3 + 8*n^2 + 4*n)*x^4)*D[P[n - 1, x], x])/(243*n^2 - 243*n + (81*n^3 - 243*n^2 + 189*n - 27)*x + (3*n^4 - 9*n^3 + 9*n^2 - 3*n)*x^2)]];
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] = ((729*n^2 - 729*n + (243*n^3 - 81*n^2 - 729*n - 81)*x + (9*n^4 + 63*n^3 - 171*n^2 - 9*n + 72)*x^2 + (2*n^4 - 6*n^2 - 4*n)*x^3)*v[i - 1] - (729*n*x + (243*n^2 + 324*n - 81)*x^2 + (9*n^3 + 126*n^2 + 9*n - 36)*x^3 + (4*n^3 + 8*n^2 + 4*n)*x^4)*deriv(v[i - 1], x))/(243*n^2 - 243*n + (81*n^3 - 243*n^2 + 189*n - 27)*x + (3*n^4 - 9*n^3 + 9*n^2 - 3*n)*x^2)); 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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.