Real-rooted atlas

A122848 → basis A013610 → basis A007318

nonnegative coefficients

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

Recurrence

\(\left(3\right)\,P_{n}(x) = \left(15 + 9\,x - 3\,n\right)\,P_{n-1}(x) + \left(8 + 6\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1437
22027956
311221613527490
468817041539594814606
5454413680160209045243024345962
631936112752161460119205473859477729482944
7236800956928161481614666407682852296353572121875311012

All 15 rows

Other recurrences

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

\(P_{n}(x) = \left(-4 - 3\,x + 4\,n + 3\,n x\right)\,P_{n-2}(x) + \left(4 + 3\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 3\,x\)
Technical details
Node id
n-fe9ce24e9dac9349 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-f6d27d927285768f)
Fingerprint
f6-eb5b4007393f32fd

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:        4,        3;
  2:       20,       27,         9;
  3:      112,      216,       135,        27;
  4:      688,     1704,      1539,       594,        81;
  5:     4544,    13680,     16020,      9045,      2430,      243;
  6:    31936,   112752,    161460,    119205,     47385,     9477,      729;
  7:   236800,   956928,   1614816,   1466640,    768285,   229635,    35721,    2187;
  8:  1841408,  8365824,  16218720,  17439408,  11303145,  4483836,  1051218,  131220,   6561;
  9: 14943232, 75292416, 164612736, 204101856, 157315284, 77631939, 24310692, 4605822, 472392, 19683;
%F Formula
The row polynomials satisfy 3*P(n,x) = -(3*n-15-9*x)*P(n-1,x) + (8+6*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[Together[(-(3*n - 15 - 9*x)*P[n - 1, x] + (8 + 6*x)*D[P[n - 1, x], x])/(3)]];
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 - 15 - 9*x)*v[i - 1] + (8 + 6*x)*deriv(v[i - 1], x))/(3)); 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