Real-rooted atlas

A075503 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-128 - 16\,x - x^{2}\right)\,P_{n-2}(x) + \left(-64\,x - 8\,x^{2}\right)\,P_{n-2}'(x) + \left(24 + 2\,x\right)\,P_{n-1}(x) + \left(8\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 24 + 2\,x\)
\(P_{1}(x) = 448 + 96\,x + 3\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8Pn(1)
024226
1448963547
276803200240411124
312697692160124804805232101
4206438424657925376003584084065104462
533292288633077762090188821504008512013447119738823
653477376015859712007638220801138851846773760177408201683005405416
7857315737639132856320268212633605573836800467496960180633603360002880980587013065

All 15 rows

Technical details
Node id
n-6b4ac16fb1963ef7 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-5319f6ff95c5881c

Observed on the stored rows ?

  • Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (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:            24,              2;
  1:           448,             96,              3;
  2:          7680,           3200,            240,              4;
  3:        126976,          92160,          12480,            480,             5;
  4:       2064384,        2465792,         537600,          35840,           840,            6;
  5:      33292288,       63307776,       20901888,        2150400,         85120,         1344,          7;
  6:     534773760,     1585971200,      763822080,      113885184,       6773760,       177408,       2016,        8;
  7:    8573157376,    39132856320,    26821263360,     5573836800,     467496960,     18063360,     336000,     2880,      9;
  8:  137304735744,   956334866432,   916979712000,   258715156480,   29407150080,   1572544512,   42577920,   591360,   3960,   10;
  9: 2197949513728, 23226646265856, 30777853083648, 11571245875200, 1734937149440, 123351072768, 4559622144, 91238400, 982080, 5280, 11;
%F Formula
The row polynomials satisfy P(n,x) = -(128+16*x+x^2)*P(n-2,x) - (64*x+8*x^2)*d/dx P(n-2,x) + (24+2*x)*P(n-1,x) + 8*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 24 + 2*x, P(1,x) = 448 + 96*x + 3*x^2.
%t Mathematica
P[0, x_] := 24 + 2*x;
P[1, x_] := 448 + 96*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[-(128 + 16*x + x^2)*P[n - 2, x] - (64*x + 8*x^2)*D[P[n - 2, x], x] + (24 + 2*x)*P[n - 1, x] + 8*x*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] = 24 + 2*x; v[2] = 448 + 96*x + 3*x^2; for(n = 2, N, my(i = n + 1); v[i] = -(128 + 16*x + x^2)*v[i - 2] - (64*x + 8*x^2)*deriv(v[i - 2], x) + (24 + 2*x)*v[i - 1] + 8*x*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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.