Real-rooted atlas

A094638 → D+1 → basis A013610

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-9\,n x^{2} + 6\,n^{2} x + 18\,n^{2} x^{2} - n^{3} - 6\,n^{3} x - 9\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-1 - 6\,n x + 2\,n^{2} + 6\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 3\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1336
213331864
370312423162967
4446295269036534194418779
532762929210050316245911939429160444084
627252310020142539333546694184784252817252488012355170
7253296351890420469096641928241159716691184547246096481211022480394847805

All 15 rows

Technical details
Node id
n-2b0d8151d6ec66ed · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-6b3534e187a8a3b2

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,         3;
  2:       13,        33,         18;
  3:       70,       312,        423,         162;
  4:      446,      2952,       6903,        6534,        1944;
  5:     3276,     29292,     100503,      162459,      119394,        29160;
  6:    27252,    310020,    1425393,     3354669,     4184784,      2528172,       524880;
  7:   253296,   3518904,   20469096,    64192824,   115971669,    118454724,     60964812,     11022480;
  8:  2602224,  42860664,  303235272,  1198186776,  2872634949,   4233119292,   3673562220,   1650537648,   264539520;
  9: 29288160, 559328976, 4677516180, 22410412020, 67484259243, 131560445187, 164244825576, 124197446772, 49590767376, 7142567040;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(n^3+(6*n^3-6*n^2)*x+(9*n^3-18*n^2+9*n)*x^2)*P(n-2,x) + (2*n^2-1+(6*n^2-6*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 3*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 3*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + (6*n^3 - 6*n^2)*x + (9*n^3 - 18*n^2 + 9*n)*x^2)*P[n - 2, x] + (2*n^2 - 1 + (6*n^2 - 6*n)*x)*P[n - 1, x])/(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 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 + (6*n^3 - 6*n^2)*x + (9*n^3 - 18*n^2 + 9*n)*x^2)*v[i - 2] + (2*n^2 - 1 + (6*n^2 - 6*n)*x)*v[i - 1])/(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

References

  1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.