Real-rooted atlas

A026729 → basis A008277 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-2 - 2\,x - x^{2}\right)\,P_{n-4}(x) + \left(-x - x^{2}\right)\,P_{n-4}'(x) + \left(-4 - 4\,x - 2\,x^{2}\right)\,P_{n-3}(x) + \left(-2\,x - 2\,x^{2}\right)\,P_{n-3}'(x) + \left(1 - x^{2}\right)\,P_{n-2}(x) + \left(-x^{2}\right)\,P_{n-2}'(x) + \left(3 + 2\,x\right)\,P_{n-1}(x) + \left(x\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 1\)
\(P_{3}(x) = 4 + 2\,x\)
\(P_{4}(x) = 13 + 16\,x + 3\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
10
211
3426
41316332
53988394170
6112416324765933
73151814220888413065357
887575441347381762010204732289
924143045476761662082416540383018204349
10663312057241793349223225003061104741342491356350
111818347117222053153451792234322078502813771112704576109425711
1249776182486411380608232177842048953090348242148146284008206197601168450930
13136151702402657787746151462444170333470961382642974335052828165457513202097912518487029
143722312691852428994675196562561213632882859651992163778339048654884011926377989880479381236134088698807
15101737410284441414418518136049086276105980906009269708466449884509212922874802280318932509882017110946958815341433508644458
Technical details
Node id
n-485e00e2d3115ebb · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-55d1360c3318f44a

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 -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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:    0;
   2:    1;
   3:    4,      2;
   4:   13,     16,      3;
   5:   39,     88,     39,      4;
   6:  112,    416,    324,     76,      5;
   7:  315,   1814,   2208,    884,    130,     6;
   8:  875,   7544,  13473,   8176,   2010,   204,    7;
   9: 2414,  30454,  76761,  66208,  24165,  4038,  301,   8;
  10: 6633, 120572, 417933, 492232, 250030, 61104, 7413, 424, 9;
%F Formula
The row polynomials satisfy P(n,x) = -(2+2*x+x^2)*P(n-4,x) - (x+x^2)*d/dx P(n-4,x) - (4+4*x+2*x^2)*P(n-3,x) - (2*x+2*x^2)*d/dx P(n-3,x) + (1-x^2)*P(n-2,x) - x^2*d/dx P(n-2,x) + (3+2*x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 5, with P(1,x) = 0, P(2,x) = 1, P(3,x) = 4 + 2*x, P(4,x) = 13 + 16*x + 3*x^2.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 1;
P[3, x_] := 4 + 2*x;
P[4, x_] := 13 + 16*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[-(2 + 2*x + x^2)*P[n - 4, x] - (x + x^2)*D[P[n - 4, x], x] - (4 + 4*x + 2*x^2)*P[n - 3, x] - (2*x + 2*x^2)*D[P[n - 3, x], x] + (1 - x^2)*P[n - 2, x] - x^2*D[P[n - 2, x], x] + (3 + 2*x)*P[n - 1, x] + x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; v[3] = 4 + 2*x; v[4] = 13 + 16*x + 3*x^2; for(n = 5, N, my(i = n); v[i] = -(2 + 2*x + x^2)*v[i - 4] - (x + x^2)*deriv(v[i - 4], x) - (4 + 4*x + 2*x^2)*v[i - 3] - (2*x + 2*x^2)*deriv(v[i - 3], x) + (1 - x^2)*v[i - 2] - x^2*deriv(v[i - 2], x) + (3 + 2*x)*v[i - 1] + x*deriv(v[i - 1], x)); 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

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.