Real-rooted atlas

basis-map coefficients of A371080, shifted → basis A013611 → basis A008277

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1549
2406816124
34401180576642260
461602300017392396825650776
51047205076405268001846402432010241349144
62094400126022401678096080156801614080138240409641249696
7481712003486380805706187203476726409501184012400640745472163841423274976

All 15 rows

Technical details
Node id
n-250be2442d9d6c70 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-fa42e2e61b188dff

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:           5,            4;
  2:          40,           68,           16;
  3:         440,         1180,          576,           64;
  4:        6160,        23000,        17392,         3968,          256;
  5:      104720,       507640,       526800,       184640,        24320,        1024;
  6:     2094400,     12602240,     16780960,      8015680,      1614080,      138240,       4096;
  7:    48171200,    348638080,    570618720,    347672640,     95011840,    12400640,     745472,     16384;
  8:  1252451200,  10651827200,  20795588800,  15494035200,   5381187840,   950476800,   86876160,   3866624,    65536;
  9: 36321084800, 356520102400, 812044094400, 718437798400, 304129593600, 68098114560, 8406343680, 567902208, 19464192, 262144;
%F Formula
The row polynomials satisfy P(n,x) = (3*n+2+4*x)*P(n-1,x) + 4*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(3*n + 2 + 4*x)*P[n - 1, x] + 4*x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (3*n + 2 + 4*x)*v[i - 1] + 4*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 - 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. [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