Real-rooted atlas

A257625 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-1 + 2\,n\right)\,P_{n}(x) = \left(9 - 54\,x - 27\,x^{2} + 18\,n - 72\,n x + 18\,n x^{2} + 72\,n^{2} x + 108\,n^{2} x^{2} - 72\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(18\,x - 18\,x^{3} + 36\,n x - 36\,n x^{2} - 72\,n^{2} x^{2} + 72\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-6 + 6\,x - 24\,n x + 24\,n^{2} x\right)\,P_{n-1}(x) + \left(-6\,x + 6\,x^{2} + 12\,n x - 12\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 6\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1369
2910827144
327124218631083240
48112312558902462440593312
52431151821226178163490428795514583265920
6729105267623057541686659683842923531580285103134369280
7218795309463979705772272056804284007100579594115433358311174966348948480

All 15 rows

Technical details
Node id
n-cb35305b0e1e69ed · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-2e37e4aadfbd49d5

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,         6;
  2:     9,       108,           27;
  3:    27,      1242,         1863,           108;
  4:    81,     12312,        55890,         24624,           405;
  5:   243,    115182,      1226178,       1634904,        287955,          1458;
  6:   729,   1052676,     23057541,      68665968,      38429235,       3158028,          5103;
  7:  2187,   9530946,    397970577,    2272056804,    2840071005,     795941154,      33358311,        17496;
  8:  6561,  85975344,   6527118996,   65223898272,  153363834270,   97835847408,   15229944324,    343901376,      59049;
  9: 19683, 774447318, 103710120660, 1709112051504, 6831309329010, 8197571194812, 2990946090132, 276560321760, 3485012931, 196830;
%F Formula
The row polynomials satisfy (2*n-1)*P(n,x) = (18*n+9+(72*n^2-72*n-54)*x-(72*n^3-108*n^2-18*n+27)*x^2)*P(n-2,x) + ((36*n+18)*x-(72*n^2+36*n)*x^2+(72*n^2-18)*x^3)*d/dx P(n-2,x) - (6-(24*n^2-24*n+6)*x)*P(n-1,x) + ((12*n-6)*x-(12*n-6)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 6*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 6*x;
P[n_, x_] := P[n, x] = Expand[Together[((18*n + 9 + (72*n^2 - 72*n - 54)*x - (72*n^3 - 108*n^2 - 18*n + 27)*x^2)*P[n - 2, x] + ((36*n + 18)*x - (72*n^2 + 36*n)*x^2 + (72*n^2 - 18)*x^3)*D[P[n - 2, x], x] - (6 - (24*n^2 - 24*n + 6)*x)*P[n - 1, x] + ((12*n - 6)*x - (12*n - 6)*x^2)*D[P[n - 1, x], x])/(2*n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = ((18*n + 9 + (72*n^2 - 72*n - 54)*x - (72*n^3 - 108*n^2 - 18*n + 27)*x^2)*v[i - 2] + ((36*n + 18)*x - (72*n^2 + 36*n)*x^2 + (72*n^2 - 18)*x^3)*deriv(v[i - 2], x) - (6 - (24*n^2 - 24*n + 6)*x)*v[i - 1] + ((12*n - 6)*x - (12*n - 6)*x^2)*deriv(v[i - 1], x))/(2*n - 1)); 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

Same fingerprint ?

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.