Real-rooted atlas

A253284 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-n - n^{2} x + n^{3} x\right)\,P_{n}(x) = \left(-n - 4\,n x - 2\,n x^{2} - n^{2} - 5\,n^{2} x - 2\,n^{2} x^{2} + 2\,n^{3} x^{2} + n^{4} x + 2\,n^{4} x^{2}\right)\,P_{n-1}(x) + \left(-2\,x - 4\,x^{2} - 2\,x^{3} - 2\,n x - 4\,n x^{2} - 2\,n x^{3} + 2\,n^{2} x^{2} + 2\,n^{2} x^{3} + 2\,n^{3} x^{2} + 2\,n^{3} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
12810
2672108186
324576216019204680
41204800324006720042000146520
5720432004536001612800226800010886405466960
65040423360635040033868800793800008382528032598720236451600
740320451584091445760677376000232848000040236134403390266880110702592011622764160

All 15 rows

Technical details
Node id
n-f118564e69a2ae6e · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-bedf7aea2d311393

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:       2,         8;
  2:       6,        72,         108;
  3:      24,       576,        2160,         1920;
  4:     120,      4800,       32400,        67200,         42000;
  5:     720,     43200,      453600,      1612800,       2268000,       1088640;
  6:    5040,    423360,     6350400,     33868800,      79380000,      83825280,       32598720;
  7:   40320,   4515840,    91445760,    677376000,    2328480000,    4023613440,     3390266880,     1107025920;
  8:  362880,  52254720,  1371686400,  13412044800,   62868960000,  156920924160,   213586813440,   149448499200,   42032390400;
  9: 3628800, 653184000, 21555072000, 268240896000, 1634592960000, 5492232345600, 10679340672000, 11955879936000, 7145506368000, 1764322560000;
%F Formula
The row polynomials satisfy -(n-(n^3-n^2)*x)*P(n,x) = -(n^2+n-(n^4-5*n^2-4*n)*x-(2*n^4+2*n^3-2*n^2-2*n)*x^2)*P(n-1,x) - ((2*n+2)*x-(2*n^3+2*n^2-4*n-4)*x^2-(2*n^3+2*n^2-2*n-2)*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 + n - (n^4 - 5*n^2 - 4*n)*x - (2*n^4 + 2*n^3 - 2*n^2 - 2*n)*x^2)*P[n - 1, x] - ((2*n + 2)*x - (2*n^3 + 2*n^2 - 4*n - 4)*x^2 - (2*n^3 + 2*n^2 - 2*n - 2)*x^3)*D[P[n - 1, x], x])/(-n + (n^3 - n^2)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(n^2 + n - (n^4 - 5*n^2 - 4*n)*x - (2*n^4 + 2*n^3 - 2*n^2 - 2*n)*x^2)*v[i - 1] - ((2*n + 2)*x - (2*n^3 + 2*n^2 - 4*n - 4)*x^2 - (2*n^3 + 2*n^2 - 2*n - 2)*x^3)*deriv(v[i - 1], x))/(-n + (n^3 - n^2)*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

Same fingerprint ?

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.