Real-rooted atlas

A257621 → D

nonnegative coefficients

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

Recurrence

\(\left(-5 + 4\,n\right)\,P_{n}(x) = \left(21 - 54\,x + 45\,x^{2} - 84\,n + 240\,n x - 236\,n x^{2} - 96\,n^{2} x + 240\,n^{2} x^{2} - 64\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(12\,x - 32\,x^{2} + 20\,x^{3} - 48\,n x + 144\,n x^{2} - 96\,n x^{3} - 64\,n^{2} x^{2} + 64\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-38 + 50\,x + 40\,n - 80\,n x + 32\,n^{2} x\right)\,P_{n-1}(x) + \left(-20\,x + 20\,x^{2} + 16\,n x - 16\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
133
2421860
3393786811260
4315617292946832430240
523631284892427338945241215831600
61710064031790128201408063580855030437425945920
7121672552215966307208115409610820130539915730035015309908107200

All 15 rows

Technical details
Node id
n-4b0b2cb99a241d3f · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-19a74063fe1c0664

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • 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:        0;
  1:        3;
  2:       42,         18;
  3:      393,        786,           81;
  4:     3156,      17292,         9468,          324;
  5:    23631,     284892,       427338,        94524,         1215;
  6:   171006,    4031790,     12820140,      8063580,       855030,         4374;
  7:  1216725,   52215966,    307208115,    409610820,    130539915,      7300350,       15309;
  8:  8584872,  640078776,   6409572552,  15565211160,  10682620920,   1920236328,    60094104,     52488;
  9: 60323739, 7573128600, 122066778708, 492299236968, 615374046210, 244133557416, 26505950100, 482589912, 177147;
%F Formula
The row polynomials satisfy (4*n-5)*P(n,x) = -(84*n-21+(96*n^2-240*n+54)*x+(64*n^3-240*n^2+236*n-45)*x^2)*P(n-2,x) - ((48*n-12)*x+(64*n^2-144*n+32)*x^2-(64*n^2-96*n+20)*x^3)*d/dx P(n-2,x) + (40*n-38+(32*n^2-80*n+50)*x)*P(n-1,x) + ((16*n-20)*x-(16*n-20)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[(-(84*n - 21 + (96*n^2 - 240*n + 54)*x + (64*n^3 - 240*n^2 + 236*n - 45)*x^2)*P[n - 2, x] - ((48*n - 12)*x + (64*n^2 - 144*n + 32)*x^2 - (64*n^2 - 96*n + 20)*x^3)*D[P[n - 2, x], x] + (40*n - 38 + (32*n^2 - 80*n + 50)*x)*P[n - 1, x] + ((16*n - 20)*x - (16*n - 20)*x^2)*D[P[n - 1, x], x])/(4*n - 5)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 3; for(n = 2, N, my(i = n + 1); v[i] = (-(84*n - 21 + (96*n^2 - 240*n + 54)*x + (64*n^3 - 240*n^2 + 236*n - 45)*x^2)*v[i - 2] - ((48*n - 12)*x + (64*n^2 - 144*n + 32)*x^2 - (64*n^2 - 96*n + 20)*x^3)*deriv(v[i - 2], x) + (40*n - 38 + (32*n^2 - 80*n + 50)*x)*v[i - 1] + ((16*n - 20)*x - (16*n - 20)*x^2)*deriv(v[i - 1], x))/(4*n - 5)); 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. [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
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.