Real-rooted atlas

A115951 → basis A008277

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1224
2820634
33217613220360
413614882116760704570
5592124323004019240406025266616
62624103712402144409520148680206649241088268
7117768674565217856794483244178401031856101640343219596688

All 15 rows

Technical details
Node id
n-445f9a43ace6155b · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-44562cdf8dd879d1

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 0)
  • 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:      1;
  1:      2,        2;
  2:      8,       20,         6;
  3:     32,      176,       132,         20;
  4:    136,     1488,      2116,        760,         70;
  5:    592,    12432,     30040,      19240,       4060,        252;
  6:   2624,   103712,    402144,     409520,     148680,      20664,       924;
  7:  11776,   867456,   5217856,    7944832,    4417840,    1031856,    101640,     3432;
  8:  53344,  7285248,  66539344,  145880224,  116662840,   40737984,   6632472,   487344,   12870;
  9: 243392, 61456064, 840376032, 2587328672, 2861522160, 1401841392, 335489616, 40243632, 2290860, 48620;
%F Formula
The row polynomials satisfy n*P(n,x) = (4*n-4+(4*n-4)*x)*P(n-2,x) + (4*n-4)*x*d/dx P(n-2,x) + (4*n-2+(4*n-2)*x)*P(n-1,x) + (4*n-2)*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((4*n - 4 + (4*n - 4)*x)*P[n - 2, x] + (4*n - 4)*x*D[P[n - 2, x], x] + (4*n - 2 + (4*n - 2)*x)*P[n - 1, x] + (4*n - 2)*x*D[P[n - 1, x], x])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((4*n - 4 + (4*n - 4)*x)*v[i - 2] + (4*n - 4)*x*deriv(v[i - 2], x) + (4*n - 2 + (4*n - 2)*x)*v[i - 1] + (4*n - 2)*x*deriv(v[i - 1], x))/(n)); 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

  • Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
    by Ray basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
    derived from Ray basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
    derived from Ray basis map theorem

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