Real-rooted atlas

A131689 → (θ+1)² → reverse

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6Pn(1)
111
244
318422
496544154
560057612641306
6432060002400270412994
735280648003900086405584148282
832256074088060480021000028896113441908274

All 15 rows

Technical details
Node id
n-b693c57370340d92 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-8b3c2acb7820153e

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
  • 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:       18,         4;
   4:       96,        54,         4;
   5:      600,       576,       126,        4;
   6:     4320,      6000,      2400,      270,        4;
   7:    35280,     64800,     39000,     8640,      558,       4;
   8:   322560,    740880,    604800,   210000,    28896,    1134,      4;
   9:  3265920,   9031680,   9384480,  4536000,  1020600,   92736,   2286,    4;
  10: 36288000, 117573120, 149022720, 93350880, 30028320, 4662000, 290400, 4590, 4;
%F Formula
The row polynomials satisfy P(n,x) = -(n-1-4*x-(n-3)*x^2)*P(n-2,x) + (x-x^3)*d/dx P(n-2,x) + (n+2+(n-3)*x)*P(n-1,x) - (x+x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 4.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 4;
P[n_, x_] := P[n, x] = Expand[-(n - 1 - 4*x - (n - 3)*x^2)*P[n - 2, x] + (x - x^3)*D[P[n - 2, x], x] + (n + 2 + (n - 3)*x)*P[n - 1, x] - (x + 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; v[2] = 4; for(n = 3, N, my(i = n); v[i] = -(n - 1 - 4*x - (n - 3)*x^2)*v[i - 2] + (x - x^3)*deriv(v[i - 2], x) + (n + 2 + (n - 3)*x)*v[i - 1] - (x + 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.