Real-rooted atlas

A160562 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-9 - 2\,x - x^{2}\right)\,P_{n-2}(x) + \left(-16\,x - 12\,x^{2}\right)\,P_{n-2}'(x) + \left(-4\,x^{2} - 4\,x^{3}\right)\,P_{n-2}''(x) + \left(10 + 2\,x\right)\,P_{n-1}(x) + \left(16\,x\right)\,P_{n-1}'(x) + \left(4\,x^{2}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
111
210212
391703164
4820193225243008
573814994015246660573232
666430126326282196473788143062226880
7597871317144104217112970727802652652730781824192

All 15 rows

Technical details
Node id
n-e5e4323c11e2a9a6 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-abe2f5030b54dabd

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:        0;
  1:        1;
  2:       10,           2;
  3:       91,          70,            3;
  4:      820,        1932,          252,           4;
  5:     7381,       49940,        15246,         660,          5;
  6:    66430,     1263262,       821964,       73788,       1430,         6;
  7:   597871,    31714410,     42171129,     7072780,     265265,      2730,       7;
  8:  5380840,   794055992,   2113956936,   629123352,   40938040,    779688,    4760,    8;
  9: 48427561, 19862161480, 104774973852, 53777600760, 5739907030, 180892920, 1980636, 7752, 9;
%F Formula
The row polynomials satisfy P(n,x) = -(9+2*x+x^2)*P(n-2,x) - (16*x+12*x^2)*d/dx P(n-2,x) - (4*x^2+4*x^3)*d^2/dx^2 P(n-2,x) + (10+2*x)*P(n-1,x) + 16*x*d/dx P(n-1,x) + 4*x^2*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(9 + 2*x + x^2)*P[n - 2, x] - (16*x + 12*x^2)*D[P[n - 2, x], x] - (4*x^2 + 4*x^3)*D[P[n - 2, x], {x, 2}] + (10 + 2*x)*P[n - 1, x] + 16*x*D[P[n - 1, x], x] + 4*x^2*D[P[n - 1, x], {x, 2}]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = -(9 + 2*x + x^2)*v[i - 2] - (16*x + 12*x^2)*deriv(v[i - 2], x) - (4*x^2 + 4*x^3)*deriv(deriv(v[i - 2], x), x) + (10 + 2*x)*v[i - 1] + 16*x*deriv(v[i - 1], x) + 4*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 - 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.