Real-rooted atlas

A156289 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-4 - 5\,x - 2\,x^{2} + 3\,x^{3}\right)\,P_{n-2}(x) + \left(-5\,x - 2\,x^{2} + 3\,x^{3} + 2\,x^{4}\right)\,P_{n-2}'(x) + \left(-x^{2} + x^{4}\right)\,P_{n-2}''(x) + \left(5 + 5\,x - x^{2}\right)\,P_{n-1}(x) + \left(5\,x + 2\,x^{2}\right)\,P_{n-1}'(x) + \left(x^{2}\right)\,P_{n-1}''(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 4 + 3\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2437
316451576
4644835251051177
5256466511655693094524451
610244326321862527373510395010395650992
74096394485379879587687606621615175675513513521479641
816384357084363533925253017765332432100168378210331080752027025856084327

All 15 rows

Technical details
Node id
n-3b22cec39202b5b9 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-e5341cf5487df687

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:     16,        45,          15;
   4:     64,       483,         525,          105;
   5:    256,      4665,       11655,         6930,          945;
   6:   1024,     43263,      218625,       273735,       103950,        10395;
   7:   4096,    394485,     3798795,      8768760,      6621615,      1756755,       135135;
   8:  16384,   3570843,    63533925,    253017765,    332432100,    168378210,     33108075,      2027025;
   9:  65536,  32219505,  1041505935,   6893694990,  14709579885,  12455933490,   4534860330,    689188500,    34459425;
  10: 262144, 290303223, 16889500425, 181683774195, 604691901450, 798289336845, 471841420050, 129636356850, 15713497800, 654729075;
%F Formula
The row polynomials satisfy P(n,x) = -(4+5*x+2*x^2-3*x^3)*P(n-2,x) - (5*x+2*x^2-3*x^3-2*x^4)*d/dx P(n-2,x) - (x^2-x^4)*d^2/dx^2 P(n-2,x) + (5+5*x-x^2)*P(n-1,x) + (5*x+2*x^2)*d/dx P(n-1,x) + x^2*d^2/dx^2 P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 4 + 3*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 4 + 3*x;
P[n_, x_] := P[n, x] = Expand[-(4 + 5*x + 2*x^2 - 3*x^3)*P[n - 2, x] - (5*x + 2*x^2 - 3*x^3 - 2*x^4)*D[P[n - 2, x], x] - (x^2 - x^4)*D[P[n - 2, x], {x, 2}] + (5 + 5*x - x^2)*P[n - 1, x] + (5*x + 2*x^2)*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}]];
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 + 3*x; for(n = 3, N, my(i = n); v[i] = -(4 + 5*x + 2*x^2 - 3*x^3)*v[i - 2] - (5*x + 2*x^2 - 3*x^3 - 2*x^4)*deriv(v[i - 2], x) - (x^2 - x^4)*deriv(deriv(v[i - 2], x), x) + (5 + 5*x - x^2)*v[i - 1] + (5*x + 2*x^2)*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], x), 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

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.