Real-rooted atlas

A059473 → γ → θ+1

nonnegative coefficients

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

Recurrence

\(\left(2\,n + n^{2} x\right)\,P_{n}(x) = \left(-8\,x - 4\,x^{2} + 4\,n + 8\,n x + 2\,n x^{2} + 2\,n^{2} x + 2\,n^{2} x^{2}\right)\,P_{n-1}(x) + \left(-4\,x - 12\,x^{2} - 8\,x^{3} - 2\,n x^{2} - 4\,n x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
122
2448
381624
416481276
53212872232
66432028832704
71287689602562112
8256179228801280806288
951240968064512080018592
10102492162150417920480019254656
112048204805529657344224002304159872
124096450561382401720328960016128448465600
138192983043379204915203225608601662721350784
1416384212992811008135168010752003870725017610243905536
Technical details
Node id
n-ad7b33c72a99c85c · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-6d1011f498340db8

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,    4;
  3:   8,   16;
  4:  16,   48,   12;
  5:  32,  128,   72;
  6:  64,  320,  288,   32;
  7: 128,  768,  960,  256;
  8: 256, 1792, 2880, 1280,  80;
  9: 512, 4096, 8064, 5120, 800;
%F Formula
The row polynomials satisfy (2*n+n^2*x)*P(n,x) = (4*n+(2*n^2+8*n-8)*x+(2*n^2+2*n-4)*x^2)*P(n-1,x) - (4*x+(2*n+12)*x^2+(4*n+8)*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[((4*n + (2*n^2 + 8*n - 8)*x + (2*n^2 + 2*n - 4)*x^2)*P[n - 1, x] - (4*x + (2*n + 12)*x^2 + (4*n + 8)*x^3)*D[P[n - 1, x], x])/(2*n + 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] = ((4*n + (2*n^2 + 8*n - 8)*x + (2*n^2 + 2*n - 4)*x^2)*v[i - 1] - (4*x + (2*n + 12)*x^2 + (4*n + 8)*x^3)*deriv(v[i - 1], x))/(2*n + 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

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.