Real-rooted atlas

A225470 → basis A008277 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-12 + 12\,x + 9\,x^{2} + 21\,n - 12\,n x - 9\,n^{2}\right)\,P_{n-2}(x) + \left(6\,x - 12\,x^{2} + 4\,x^{3} - 6\,n x + 12\,n x^{2}\right)\,P_{n-2}'(x) + \left(-x^{2} + 4\,x^{3}\right)\,P_{n-2}''(x) + \left(-4 + 4\,x + 6\,n\right)\,P_{n-1}(x) + \left(2\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1347
21840967
316247218916839
41944678437085762513037
5291601163207830016960137536242151
652488023267201827720495360567502808495234287
7110224805328736047272680151379202192750154728514564129073127

All 15 rows

Technical details
Node id
n-ea8a0d1e4415b9ae · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-61b2e0e429020a42

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,           4;
  2:         18,          40,           9;
  3:        162,         472,         189,          16;
  4:       1944,        6784,        3708,         576,         25;
  5:      29160,      116320,       78300,       16960,       1375,        36;
  6:     524880,     2326720,     1827720,      495360,      56750,      2808,       49;
  7:   11022480,    53287360,    47272680,    15137920,    2192750,    154728,     5145,     64;
  8:  264539520,  1376273920,  1348986240,   492764160,   85050000,   7644672,   364952,   8704,    81;
  9: 7142567040, 39593827840, 42217217280, 17181122560, 3406494000, 367101504, 22448664, 772608, 13851, 100;
%F Formula
The row polynomials satisfy P(n,x) = -(9*n^2-21*n+12+(12*n-12)*x-9*x^2)*P(n-2,x) - ((6*n-6)*x-(12*n-12)*x^2-4*x^3)*d/dx P(n-2,x) - (x^2-4*x^3)*d^2/dx^2 P(n-2,x) + (6*n-4+4*x)*P(n-1,x) + (2*x-4*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 4*x;
P[n_, x_] := P[n, x] = Expand[-(9*n^2 - 21*n + 12 + (12*n - 12)*x - 9*x^2)*P[n - 2, x] - ((6*n - 6)*x - (12*n - 12)*x^2 - 4*x^3)*D[P[n - 2, x], x] - (x^2 - 4*x^3)*D[P[n - 2, x], {x, 2}] + (6*n - 4 + 4*x)*P[n - 1, x] + (2*x - 4*x^2)*D[P[n - 1, x], 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 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(9*n^2 - 21*n + 12 + (12*n - 12)*x - 9*x^2)*v[i - 2] - ((6*n - 6)*x - (12*n - 12)*x^2 - 4*x^3)*deriv(v[i - 2], x) - (x^2 - 4*x^3)*deriv(deriv(v[i - 2], x), x) + (6*n - 4 + 4*x)*v[i - 1] + (2*x - 4*x^2)*deriv(v[i - 1], 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. [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. [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
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.