Real-rooted atlas

A026729 → basis A013610 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(-3 + 3\,n\right)\,P_{n}(x) = \left(24 + 9\,x - 6\,n\right)\,P_{n-1}(x) + \left(32 + 36\,x + 9\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2437
32027956
49619212627441
546412241197513813479
62240734495406129194424327440
710816423846865258779280267047729216433
8522242380804619524942083146041190702478621871707111

All 15 rows

Other recurrences

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

\(P_{n}(x) = \left(4 + 3\,x\right)\,P_{n-2}(x) + \left(4 + 3\,x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 4 + 3\,x\)
Technical details
Node id
n-e086e6a4622a1cba · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-17e03e7ad1a826ee)
Fingerprint
f6-e13f244cbbb9f952

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:       4,       3;
   3:      20,      27,        9;
   4:      96,     192,      126,       27;
   5:     464,    1224,     1197,      513,       81;
   6:    2240,    7344,     9540,     6129,     1944,      243;
   7:   10816,   42384,    68652,    58779,    28026,     7047,     729;
   8:   52224,  238080,   461952,   494208,   314604,   119070,   24786,    2187;
   9:  252160, 1310976,  2963808,  3803760,  3029481,  1532358,  480411,   85293,   6561;
  10: 1217536, 7109376, 18350208, 27469152, 26270244, 16637967, 6975072, 1865511, 288684, 19683;
%F Formula
The row polynomials satisfy (3*n-3)*P(n,x) = -(6*n-24-9*x)*P(n-1,x) + (32+36*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[(-(6*n - 24 - 9*x)*P[n - 1, x] + (32 + 36*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] = (-(6*n - 24 - 9*x)*v[i - 1] + (32 + 36*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