Real-rooted atlas

A128099 → basis A013611 → D

nonnegative coefficients

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

Recurrence

\(\left(-8 + 4\,n\right)\,P_{n}(x) = \left(-52 - 192\,x + 20\,n + 64\,n x\right)\,P_{n-1}(x) + \left(-9 - 68\,x - 128\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 8\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
10
288
31616
456128184
5128384512
6328153615363400
77524352614411248
8175212672276481638458456
93936334089216081920211424
10880886784299520409600163840968552
111940821657688473615564809830403660240
1242488531072252518455705605406720157286415648888
13922241274752689356818120704229376001101004860328896
1419904830167041831680056197120894566406606028814680064247926664
Technical details
Node id
n-9b5125b961f94faf · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-a142121d5e86b189

Observed on the stored rows ?

  • Degrees: -1, -1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6 (half affine: slope 1, intercept -2)
  • 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:    0;
  2:    8;
  3:   16;
  4:   56,   128;
  5:  128,   384;
  6:  328,  1536,  1536;
  7:  752,  4352,  6144;
  8: 1752, 12672, 27648, 16384;
  9: 3936, 33408, 92160, 81920;
%F Formula
The row polynomials satisfy (4*n-8)*P(n,x) = (20*n-52+(64*n-192)*x)*P(n-1,x) - (9+68*x+128*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 8.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 8;
P[n_, x_] := P[n, x] = Expand[Together[((20*n - 52 + (64*n - 192)*x)*P[n - 1, x] - (9 + 68*x + 128*x^2)*D[P[n - 1, x], x])/(4*n - 8)]];
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] = 0; v[3] = 8; for(n = 3, N, my(i = n + 1); v[i] = ((20*n - 52 + (64*n - 192)*x)*v[i - 1] - (9 + 68*x + 128*x^2)*deriv(v[i - 1], x))/(4*n - 8)); 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.