Real-rooted atlas

A026729 → basis A013611 → reverse

nonnegative coefficients

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

Recurrence

\(\left(-4 + 4\,n\right)\,P_{n}(x) = \left(-16 - 28\,x - 10\,x^{2} + 16\,n + 16\,n x + 5\,n x^{2}\right)\,P_{n-1}(x) + \left(-16\,x - 24\,x^{2} - 5\,x^{3}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2415
31612230
46480283175
52564482566051025
610242304185668812086000
740961126411776608016642321335125
81638453248686084582417344376043621205625
965536245760376832310272151296451208096804341203750
102621441114112198246419456001167360446976109888168161460557046875
111048576498073610092544115179528232960387072012183042546563396826208941253125
1241943042202009650069504652083205421465630330880116992003123456567680671044656144241500000
131677721696468992243269632356253696338231296219987968100843520329369607631104122662413017682082331413765625
1467108864419430400116182220818905825282020081664150025011279890022430971904087654400179317762583424248736143723778276328125
15268435456181193932854693724169793699840116391936009730260992591403417626611384328929280002228428804080307253256964692322502061048450468750
Technical details
Node id
n-304dfb3caed26af2 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-b3d4d0fb6f0d839f

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:      1;
   2:      4,       1;
   3:     16,      12,       2;
   4:     64,      80,      28,       3;
   5:    256,     448,     256,      60,       5;
   6:   1024,    2304,    1856,     688,     120,      8;
   7:   4096,   11264,   11776,    6080,    1664,    232,     13;
   8:  16384,   53248,   68608,   45824,   17344,   3760,    436,    21;
   9:  65536,  245760,  376832,  310272,  151296,  45120,   8096,   804,   34;
  10: 262144, 1114112, 1982464, 1945600, 1167360, 446976, 109888, 16816, 1460, 55;
%F Formula
The row polynomials satisfy (4*n-4)*P(n,x) = (16*n-16+(16*n-28)*x+(5*n-10)*x^2)*P(n-1,x) - (16*x+24*x^2+5*x^3)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((16*n - 16 + (16*n - 28)*x + (5*n - 10)*x^2)*P[n - 1, x] - (16*x + 24*x^2 + 5*x^3)*D[P[n - 1, x], x])/(4*n - 4)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((16*n - 16 + (16*n - 28)*x + (5*n - 10)*x^2)*v[i - 1] - (16*x + 24*x^2 + 5*x^3)*deriv(v[i - 1], x))/(4*n - 4)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

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