Real-rooted atlas

A257607 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(4 + n\right)\,P_{n}(x) = \left(-100 - 120\,x - 60\,x^{2} - 20\,n - 64\,n x - 47\,n x^{2} - 8\,n^{2} x - 12\,n^{2} x^{2} - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-20\,x + 20\,x^{3} - 4\,n x - 5\,n x^{2} + 9\,n x^{3} - n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(40 + 32\,x + 9\,n + 16\,n x + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(4\,x - 4\,x^{2} + n x - n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 5 + 10\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
151015
22512075220
3125107016055003300
462584202247016840312551480
531256177025833034444015442518750840840
615625433120264220555110404403675129936010937514414400
77812529424702499223575794780947434754998447010298645625000259459200

All 15 rows

Technical details
Node id
n-994b7fb19b3cbd3e · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-df7467a067c1cbae

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 0)
  • 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:       1;
  1:       5,        10;
  2:      25,       120,         75;
  3:     125,      1070,       1605,         500;
  4:     625,      8420,      22470,       16840,        3125;
  5:    3125,     61770,     258330,      344440,      154425,       18750;
  6:   15625,    433120,    2642205,     5511040,     4403675,     1299360,      109375;
  7:   78125,   2942470,   24992235,    75794780,    94743475,    49984470,    10298645,     625000;
  8:  390625,  19529820,  223496400,   939588040,  1705382550,  1409382060,   521491600,   78119280,   3515625;
  9: 1953125, 127335170, 1916011560, 10794651520, 27093293450, 32511952140, 18890640160, 5109364160, 573008265, 19531250;
%F Formula
The row polynomials satisfy (n+4)*P(n,x) = -(20*n+100+(8*n^2+64*n+120)*x+(n^3+12*n^2+47*n+60)*x^2)*P(n-2,x) - ((4*n+20)*x+(n^2+5*n)*x^2-(n^2+9*n+20)*x^3)*d/dx P(n-2,x) + (9*n+40+(2*n^2+16*n+32)*x)*P(n-1,x) + ((n+4)*x-(n+4)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + 10*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 5 + 10*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(20*n + 100 + (8*n^2 + 64*n + 120)*x + (n^3 + 12*n^2 + 47*n + 60)*x^2)*P[n - 2, x] - ((4*n + 20)*x + (n^2 + 5*n)*x^2 - (n^2 + 9*n + 20)*x^3)*D[P[n - 2, x], x] + (9*n + 40 + (2*n^2 + 16*n + 32)*x)*P[n - 1, x] + ((n + 4)*x - (n + 4)*x^2)*D[P[n - 1, x], x])/(n + 4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 5 + 10*x; for(n = 2, N, my(i = n + 1); v[i] = (-(20*n + 100 + (8*n^2 + 64*n + 120)*x + (n^3 + 12*n^2 + 47*n + 60)*x^2)*v[i - 2] - ((4*n + 20)*x + (n^2 + 5*n)*x^2 - (n^2 + 9*n + 20)*x^3)*deriv(v[i - 2], x) + (9*n + 40 + (2*n^2 + 16*n + 32)*x)*v[i - 1] + ((n + 4)*x - (n + 4)*x^2)*deriv(v[i - 1], x))/(n + 4)); 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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.