Real-rooted atlas

A099040 → basis A013609 → basis A007318

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(8 + 4\,x - 2\,n\right)\,P_{n-1}(x) + \left(15 + 16\,x + 4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
16410
2425216110
3288528320641200
4198048004336172825613100
51360841040492482937687041024143000
6935283373925048644009601781764198440961561000
7642816269913648384004798464284262410055681966081638417040000

All 15 rows

Other recurrences

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

\(P_{n}(x) = \left(6 + 4\,x\right)\,P_{n-2}(x) + \left(6 + 4\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 6 + 4\,x\)
Technical details
Node id
n-ce6e7d197c1a28a4 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-ad86a6e4b4c578ef)
Fingerprint
f6-b459081d61ad1ccc

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:        6,         4;
  2:       42,        52,        16;
  3:      288,       528,       320,        64;
  4:     1980,      4800,      4336,      1728,       256;
  5:    13608,     41040,     49248,     29376,      8704,      1024;
  6:    93528,    337392,    504864,    400960,    178176,     41984,      4096;
  7:   642816,   2699136,   4838400,   4798464,   2842624,   1005568,    196608,    16384;
  8:  4418064,  21164544,  44205696,  52569600,  38922496,  18368512,   5394432,   901120,   65536;
  9: 30365280, 163425600, 389719296, 540384768, 480062976, 283304960, 111042560, 27869184, 4063232, 262144;
%F Formula
The row polynomials satisfy n*P(n,x) = -(2*n-8-4*x)*P(n-1,x) + (15+16*x+4*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 - 8 - 4*x)*P[n - 1, x] + (15 + 16*x + 4*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 - 8 - 4*x)*v[i - 1] + (15 + 16*x + 4*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

Same fingerprint ?

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