Real-rooted atlas

A008292 → γ (scale 1/4) (shift +1) → x ↦ x+1 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(8\,x - 2\,n + 8\,n x + 2\,n x^{2} - 3\,n^{2} - 2\,n^{2} x + n^{2} x^{2} - n^{3} - 2\,n^{3} x - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(4\,x + 4\,x^{2} + 6\,n x + 8\,n x^{2} + 2\,n x^{3} + 2\,n^{2} x + 4\,n^{2} x^{2} + 2\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(2 + 4\,n - 2\,n x + 2\,n^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(-2\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3Pn(1)
011
122
26410
3243256
412024048408
572019208163456
650401680011088108834016
74032016128014515231744378496

All 15 rows

Technical details
Node id
n-cea077711a66bafa · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-b5dabfc3060ee3f6

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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;
  2:       6,        4;
  3:      24,       32;
  4:     120,      240,       48;
  5:     720,     1920,      816;
  6:    5040,    16800,    11088,     1088;
  7:   40320,   161280,   145152,    31744;
  8:  362880,  1693440,  1935360,   675840,   39680;
  9: 3628800, 19353600, 26853120, 13025280, 1768960;
%F Formula
The row polynomials satisfy n*P(n,x) = -(n^3+3*n^2+2*n+(2*n^3+2*n^2-8*n-8)*x+(n^3-n^2-2*n)*x^2)*P(n-2,x) + ((2*n^2+6*n+4)*x+(4*n^2+8*n+4)*x^2+(2*n^2+2*n)*x^3)*d/dx P(n-2,x) + (2*n^2+4*n+2+(2*n^2-2*n)*x)*P(n-1,x) - (2*n*x+2*n*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + 3*n^2 + 2*n + (2*n^3 + 2*n^2 - 8*n - 8)*x + (n^3 - n^2 - 2*n)*x^2)*P[n - 2, x] + ((2*n^2 + 6*n + 4)*x + (4*n^2 + 8*n + 4)*x^2 + (2*n^2 + 2*n)*x^3)*D[P[n - 2, x], x] + (2*n^2 + 4*n + 2 + (2*n^2 - 2*n)*x)*P[n - 1, x] - (2*n*x + 2*n*x^2)*D[P[n - 1, x], x])/(n)]];
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] = 2; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 + 3*n^2 + 2*n + (2*n^3 + 2*n^2 - 8*n - 8)*x + (n^3 - n^2 - 2*n)*x^2)*v[i - 2] + ((2*n^2 + 6*n + 4)*x + (4*n^2 + 8*n + 4)*x^2 + (2*n^2 + 2*n)*x^3)*deriv(v[i - 2], x) + (2*n^2 + 4*n + 2 + (2*n^2 - 2*n)*x)*v[i - 1] - (2*n*x + 2*n*x^2)*deriv(v[i - 1], x))/(n)); 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.