Real-rooted atlas

A099093 → reverse → θ+1

nonnegative coefficients

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

Recurrence

\(\left(3\,n + n^{2} x\right)\,P_{n}(x) = \left(-12\,x - 4\,x^{2} + 9\,n + 12\,n x + 2\,n x^{2} + 3\,n^{2} x + 2\,n^{2} x^{2}\right)\,P_{n-1}(x) + \left(-9\,x - 18\,x^{2} - 8\,x^{3} - 3\,n x^{2} - 4\,n x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
133
29615
3273663
48116227270
52436482431134
6729243014581084725
7218787487290129619521
865613061832805972040580109
919683104976137781583206075326835
10590493542945511243061805467514581326780
11177147118098021257641469664382725262445362524
125314413897234797161566134882296350275562510321590793
13159432312754584292292552834352012400290220449610716386633631
144782969414523981052253181169170206200145014880348128595617496346562955
Technical details
Node id
n-730ec757d728ff61 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-96951f4c8e461c2c

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:     9,      6;
  3:    27,     36;
  4:    81,    162,     27;
  5:   243,    648,    243;
  6:   729,   2430,   1458,   108;
  7:  2187,   8748,   7290,  1296;
  8:  6561,  30618,  32805,  9720,  405;
  9: 19683, 104976, 137781, 58320, 6075;
%F Formula
The row polynomials satisfy (3*n+n^2*x)*P(n,x) = (9*n+(3*n^2+12*n-12)*x+(2*n^2+2*n-4)*x^2)*P(n-1,x) - (9*x+(3*n+18)*x^2+(4*n+8)*x^3)*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[((9*n + (3*n^2 + 12*n - 12)*x + (2*n^2 + 2*n - 4)*x^2)*P[n - 1, x] - (9*x + (3*n + 18)*x^2 + (4*n + 8)*x^3)*D[P[n - 1, x], x])/(3*n + n^2*x)]];
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] = ((9*n + (3*n^2 + 12*n - 12)*x + (2*n^2 + 2*n - 4)*x^2)*v[i - 1] - (9*x + (3*n + 18)*x^2 + (4*n + 8)*x^3)*deriv(v[i - 1], x))/(3*n + n^2*x)); 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. [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.