Real-rooted atlas

A026729 → basis A013609 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(-2 + 2\,n\right)\,P_{n}(x) = \left(14 + 4\,x - 4\,n\right)\,P_{n-1}(x) + \left(21 + 20\,x + 4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2325
31214430
44578448175
5171390328120161025
6648183620521128304326000
72457831611592850434567366435125
893153666661236561843054498241728128205625

All 15 rows

Other recurrences

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

\(P_{n}(x) = \left(3 + 2\,x\right)\,P_{n-2}(x) + \left(3 + 2\,x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 3 + 2\,x\)
Technical details
Node id
n-40465177841ad1ac · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-9061d9e71b9fb393)
Fingerprint
f6-460da53f4102c01c

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:      3,      2;
   3:     12,     14,       4;
   4:     45,     78,      44,       8;
   5:    171,    390,     328,     120,      16;
   6:    648,   1836,    2052,    1128,     304,     32;
   7:   2457,   8316,   11592,    8504,    3456,    736,     64;
   8:   9315,  36666,   61236,   56184,   30544,   9824,   1728,   128;
   9:  35316, 158490,  308448,  339720,  231376,  99680,  26496,  3968,  256;
  10: 133893, 674730, 1499364, 1927080, 1577568, 852352, 303680, 68736, 8960, 512;
%F Formula
The row polynomials satisfy (2*n-2)*P(n,x) = -(4*n-14-4*x)*P(n-1,x) + (21+20*x+4*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[(-(4*n - 14 - 4*x)*P[n - 1, x] + (21 + 20*x + 4*x^2)*D[P[n - 1, x], x])/(2*n - 2)]];
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] = (-(4*n - 14 - 4*x)*v[i - 1] + (21 + 20*x + 4*x^2)*deriv(v[i - 1], x))/(2*n - 2)); 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