Real-rooted atlas

A026729 → basis A008277 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(2 + x\right)\,P_{n-2}(x) + \left(1 + x\right)\,P_{n-2}'(x) + \left(2 + x\right)\,P_{n-1}(x) + \left(1 + x\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 2 + x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2213
376114
4253211169
510217089171379
64569336561942412264
7221953574716190036632114591
8116403228934060174024599627411100659

All 15 rows

Technical details
Node id
n-5850d1b617b606a6 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-af694739d2ce3984

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:      2,       1;
   3:      7,       6,       1;
   4:     25,      32,      11,       1;
   5:    102,     170,      89,      17,      1;
   6:    456,     933,     656,     194,     24,      1;
   7:   2219,    5357,    4716,    1900,    366,     32,     1;
   8:  11640,   32289,   34060,   17402,   4599,    627,    41,    1;
   9:  65364,  204349,  250656,  155146,  52387,   9830,  1002,   51,  1;
  10: 390646, 1356350, 1893146, 1375400, 566749, 136443, 19165, 1519, 62, 1;
%F Formula
The row polynomials satisfy P(n,x) = (2+x)*P(n-2,x) + (1+x)*d/dx P(n-2,x) + (2+x)*P(n-1,x) + (1+x)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[(2 + x)*P[n - 2, x] + (1 + x)*D[P[n - 2, x], x] + (2 + x)*P[n - 1, x] + (1 + x)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + x; for(n = 3, N, my(i = n); v[i] = (2 + x)*v[i - 2] + (1 + x)*deriv(v[i - 2], x) + (2 + x)*v[i - 1] + (1 + x)*deriv(v[i - 1], x)); 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