Real-rooted atlas

A099040 → basis A013610 → basis A007318

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(10 + 6\,x - 2\,n\right)\,P_{n-1}(x) + \left(16 + 20\,x + 6\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
18614
27210236210
364013449362163136
4569615840164527560129646844
550688175488242208166536570247776699720
645107218689283217248294472815111364121284665610451896
7401408019365888399421444564684831212864127681922892672279936156122624

All 15 rows

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(P_{n}(x) = \left(8 + 6\,x\right)\,P_{n-2}(x) + \left(8 + 6\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 8 + 6\,x\)
Technical details
Node id
n-6c844b0e40bf4f2b · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-ddfbccabbc6b70d5)
Fingerprint
f6-2d35f0238a1794fe

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:         8,          6;
  2:        72,        102,         36;
  3:       640,       1344,        936,        216;
  4:      5696,      15840,      16452,       7560,       1296;
  5:     50688,     175488,     242208,     166536,      57024,       7776;
  6:    451072,    1868928,    3217248,    2944728,    1511136,     412128,      46656;
  7:   4014080,   19365888,   39942144,   45646848,   31212864,   12768192,    2892672,    279936;
  8:  35721216,  196669440,  472684032,  647688960,  553341456,  301786560,  102596544,  19875456,   1679616;
  9: 317882368, 1966694400, 5397221376, 8622443520, 8836449408, 6023763936, 2731242240, 794178432, 134369280, 10077696;
%F Formula
The row polynomials satisfy n*P(n,x) = -(2*n-10-6*x)*P(n-1,x) + (16+20*x+6*x^2)*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[Together[(-(2*n - 10 - 6*x)*P[n - 1, x] + (16 + 20*x + 6*x^2)*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; for(n = 1, N, my(i = n + 1); v[i] = (-(2*n - 10 - 6*x)*v[i - 1] + (16 + 20*x + 6*x^2)*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

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