Real-rooted atlas

A059473 → γ → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4\,n - 4\,n x + 4\,n^{2} + 6\,n^{3} x + n^{3} x^{2} + n^{4} x^{2}\right)\,P_{n}(x) = \left(-32\,x - 8\,x^{2} - 8\,x^{3} + 8\,n - 8\,n x - 28\,n x^{2} - 8\,n x^{3} + 8\,n^{2} + 32\,n^{2} x + 16\,n^{2} x^{2} + 6\,n^{2} x^{3} + 12\,n^{3} x + 22\,n^{3} x^{2} + 8\,n^{3} x^{3} + 2\,n^{4} x^{2} + 2\,n^{4} x^{3}\right)\,P_{n-1}(x) + \left(-8\,x - 24\,x^{2} - 24\,x^{3} - 16\,x^{4} - 8\,n x - 40\,n x^{2} - 64\,n x^{3} - 32\,n x^{4} - 12\,n^{2} x^{2} - 34\,n^{2} x^{3} - 20\,n^{2} x^{4} - 2\,n^{3} x^{3} - 4\,n^{3} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
122
24812
383240
4169636148
532256216504
6646408641281696
71281536288010245568
825635848640512040018000
951281922419220480400057376
101024184326451271680240001152180800
1120484096016588822937611200013824564096
124096901124147206881284480009676831361744960
138192196608101376019660801612800516096439045357440
14163844259842433024540672053760002322432351232819216339968
Technical details
Node id
n-4a58db838ece5aa9 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-b9193847b07222ca

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:   4,    8;
  3:   8,   32;
  4:  16,   96,    36;
  5:  32,  256,   216;
  6:  64,  640,   864,   128;
  7: 128, 1536,  2880,  1024;
  8: 256, 3584,  8640,  5120,  400;
  9: 512, 8192, 24192, 20480, 4000;
%F Formula
The row polynomials satisfy (4*n^2+4*n+(6*n^3-4*n)*x+(n^4+n^3)*x^2)*P(n,x) = (8*n^2+8*n+(12*n^3+32*n^2-8*n-32)*x+(2*n^4+22*n^3+16*n^2-28*n-8)*x^2+(2*n^4+8*n^3+6*n^2-8*n-8)*x^3)*P(n-1,x) - ((8*n+8)*x+(12*n^2+40*n+24)*x^2+(2*n^3+34*n^2+64*n+24)*x^3+(4*n^3+20*n^2+32*n+16)*x^4)*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[((8*n^2 + 8*n + (12*n^3 + 32*n^2 - 8*n - 32)*x + (2*n^4 + 22*n^3 + 16*n^2 - 28*n - 8)*x^2 + (2*n^4 + 8*n^3 + 6*n^2 - 8*n - 8)*x^3)*P[n - 1, x] - ((8*n + 8)*x + (12*n^2 + 40*n + 24)*x^2 + (2*n^3 + 34*n^2 + 64*n + 24)*x^3 + (4*n^3 + 20*n^2 + 32*n + 16)*x^4)*D[P[n - 1, x], x])/(4*n^2 + 4*n + (6*n^3 - 4*n)*x + (n^4 + n^3)*x^2)]];
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] = ((8*n^2 + 8*n + (12*n^3 + 32*n^2 - 8*n - 32)*x + (2*n^4 + 22*n^3 + 16*n^2 - 28*n - 8)*x^2 + (2*n^4 + 8*n^3 + 6*n^2 - 8*n - 8)*x^3)*v[i - 1] - ((8*n + 8)*x + (12*n^2 + 40*n + 24)*x^2 + (2*n^3 + 34*n^2 + 64*n + 24)*x^3 + (4*n^3 + 20*n^2 + 32*n + 16)*x^4)*deriv(v[i - 1], x))/(4*n^2 + 4*n + (6*n^3 - 4*n)*x + (n^4 + n^3)*x^2)); 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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.