Real-rooted atlas

A099093 → basis A013609 → D

nonnegative coefficients

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

Recurrence

\(\left(6 + 12\,x - 8\,n - 12\,n x + 2\,n^{2}\right)\,P_{n}(x) = \left(-90 - 228\,x - 144\,x^{2} + 46\,n + 60\,n x - 4\,n^{2}\right)\,P_{n-1}(x) + \left(-28 - 122\,x - 168\,x^{2} - 72\,x^{3} + 7\,n + 20\,n x + 12\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 6\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
166
24272114
32347926481674
4117059049720518421978
5550836936913689849638880271188
62494820865668882411197448942402799363216348
7109998110224845509049896256119361607558272195955237113390

All 15 rows

Technical details
Node id
n-950cd9cd3493d151 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-936fd1e4a33da718

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:       6;
  2:      42,       72;
  3:     234,      792,       648;
  4:    1170,     5904,      9720,      5184;
  5:    5508,    36936,     91368,     98496,      38880;
  6:   24948,   208656,    688824,   1119744,     894240,     279936;
  7:  109998,  1102248,   4550904,   9896256,   11936160,    7558272,    1959552;
  8:  475470,  5552064,  27517320,  74965824,  121111200,  115893504,   60746112,  13436928;
  9: 2024190, 26988552, 156093480, 511132032, 1035607680, 1328296320, 1052279424, 470292480, 90699264;
%F Formula
The row polynomials satisfy (2*n^2-8*n+6-(12*n-12)*x)*P(n,x) = -(4*n^2-46*n+90-(60*n-228)*x+144*x^2)*P(n-1,x) + (7*n-28+(20*n-122)*x+(12*n-168)*x^2-72*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 6.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 6;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^2 - 46*n + 90 - (60*n - 228)*x + 144*x^2)*P[n - 1, x] + (7*n - 28 + (20*n - 122)*x + (12*n - 168)*x^2 - 72*x^3)*D[P[n - 1, x], x])/(2*n^2 - 8*n + 6 - (12*n - 12)*x)]];
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] = 6; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n^2 - 46*n + 90 - (60*n - 228)*x + 144*x^2)*v[i - 1] + (7*n - 28 + (20*n - 122)*x + (12*n - 168)*x^2 - 72*x^3)*deriv(v[i - 1], x))/(2*n^2 - 8*n + 6 - (12*n - 12)*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. [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. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.