Real-rooted atlas

A099097 → basis A007318 → basis A007318

nonnegative coefficients

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

Recurrence

\(\left(-3 + 3\,n\right)\,P_{n}(x) = \left(22 + 9\,x - 2\,n\right)\,P_{n-1}(x) + \left(44 + 40\,x + 9\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2639
33837984
424034816827783
5151629202107675817299
69576230042208610593253824368040
760488174108208662133273478449153729634257
83820801281696184147214688967025522014743207621875912433

All 15 rows

Other recurrences

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

\(P_{n}(x) = \left(2 + x\right)\,P_{n-2}(x) + \left(6 + 3\,x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 6 + 3\,x\)
Technical details
Node id
n-f9d10c14bd192922 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-14b733543ff6340f)
Fingerprint
f6-616c3950683466a7

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 -1)
  • 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:        1;
   2:        6,        3;
   3:       38,       37,         9;
   4:      240,      348,       168,        27;
   5:     1516,     2920,      2107,       675,        81;
   6:     9576,    23004,     22086,     10593,      2538,      243;
   7:    60488,   174108,    208662,    133273,     47844,     9153,      729;
   8:   382080,  1281696,   1841472,   1468896,    702552,   201474,    32076,    2187;
   9:  2413456,  9245120,  15485352,  14813000,   8850961,  3382650,   807489,  110079,   6561;
  10: 15244896, 65656560, 125612112, 140113320, 100418766, 47954283, 15258510, 3119391, 371790, 19683;
%F Formula
The row polynomials satisfy (3*n-3)*P(n,x) = -(2*n-22-9*x)*P(n-1,x) + (44+40*x+9*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n - 22 - 9*x)*P[n - 1, x] + (44 + 40*x + 9*x^2)*D[P[n - 1, x], x])/(3*n - 3)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(2*n - 22 - 9*x)*v[i - 1] + (44 + 40*x + 9*x^2)*deriv(v[i - 1], x))/(3*n - 3)); 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