Real-rooted atlas

A225473 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-4 - 5\,x + 4\,n + 4\,n x\right)\,P_{n-1}(x) + \left(-4\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2437
33240981
4384672316271399
5614413824992023208132289
612288033792032640012768016564243931687
729491209584640116428806428160153699211692072932259441
8825753603096576004533043203240652801148663041784227282135621871303134679

All 15 rows

Technical details
Node id
n-c7c3b28456d49b6f · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-efc167c4cad578f4

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:           1;
   2:           4,            3;
   3:          32,           40,            9;
   4:         384,          672,          316,           27;
   5:        6144,        13824,         9920,         2320,           81;
   6:      122880,       337920,       326400,       127680,        16564,          243;
   7:     2949120,      9584640,     11642880,      6428160,      1536992,       116920,         729;
   8:    82575360,    309657600,    453304320,    324065280,    114866304,     17842272,      821356,       2187;
   9:  2642411520,  11230248960,  19240058880,  16907304960,   7995101184,   1937101824,   202835840,    5758240,     6561;
  10: 95126814720, 451852369920, 886859366400, 925256908800, 548770037760, 182900551680, 31490557440, 2277260160, 40333924, 19683;
%F Formula
The row polynomials satisfy P(n,x) = (4*n-4+(4*n-5)*x)*P(n-1,x) - (4*x+4*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(4*n - 4 + (4*n - 5)*x)*P[n - 1, x] - (4*x + 4*x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (4*n - 4 + (4*n - 5)*x)*v[i - 1] - (4*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, v[i])))};
T(10)

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. [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.