Real-rooted atlas

A136124 → basis A013610 → θ+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1369
212422781
360282324108774
43602052321319444057974
5252016524317252646010125145888812
6201601473123281583385801802254811451031067652
7181440144676836163264359852289170010818362143261749613809744

All 15 rows

Technical details
Node id
n-50475f3c9faa4edc · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-38b9e01bfeccb468

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:       12,        42,        27;
  3:       60,       282,       324,       108;
  4:      360,      2052,      3213,      1944,       405;
  5:     2520,     16524,     31725,     26460,     10125,      1458;
  6:    20160,    147312,    328158,    338580,    180225,     48114,      5103;
  7:   181440,   1446768,   3616326,   4359852,   2891700,   1081836,    214326,    17496;
  8:  1814400,  15556320,  42673716,  58063824,  45266445,  21228480,   5929686,   909792,   59049;
  9: 19958400, 182005920, 539414316, 809396928, 715670235, 396472482, 139526226, 30338064, 3720087, 196830;
%F Formula
The row polynomials satisfy P(n,x) = -(n^2+2*n+1+(6*n+6)*x+9*x^2)*P(n-2,x) + (2*n+3+6*x)*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[-(n^2 + 2*n + 1 + (6*n + 6)*x + 9*x^2)*P[n - 2, x] + (2*n + 3 + 6*x)*P[n - 1, x]];
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] = -(n^2 + 2*n + 1 + (6*n + 6)*x + 9*x^2)*v[i - 2] + (2*n + 3 + 6*x)*v[i - 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. [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.