Real-rooted atlas

A081579 → basis A013610 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(9\,n\right)\,P_{n}(x) = \left(153 + 54\,x - 153\,n - 27\,n x + 45\,n^{2}\right)\,P_{n-1}(x) + \left(735 + 693\,x + 108\,x^{2} - 267\,n - 234\,n x - 27\,n x^{2}\right)\,P_{n-1}'(x) + \left(356 + 579\,x + 270\,x^{2} + 27\,x^{3}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1538
24142992
328544120727960
42081422430068648110256
5149653783936306162543321243108928
610812132664639383124008477031121507291158592
7780045274259739840573077649135359133558343011218712318720

All 15 rows

Technical details
Node id
n-245672ded2cc5da7 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-c3ec29745baaec8f

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:        5,         3;
  2:       41,        42,         9;
  3:      285,       441,       207,        27;
  4:     2081,      4224,      3006,       864,        81;
  5:    14965,     37839,     36306,     16254,      3321,       243;
  6:   108121,    326646,    393831,    240084,     77031,     12150,      729;
  7:   780045,   2742597,   3984057,   3077649,   1353591,    335583,    43011,    2187;
  8:  5630161,  22576908,  38369124,  35907732,  20114406,   6857460,  1379268,  148716,   6561;
  9: 40631525, 183017115, 356232420, 391697100, 266884470, 116242938, 32183892, 5432508, 505197, 19683;
%F Formula
The row polynomials satisfy 9*n*P(n,x) = (45*n^2-153*n+153-(27*n-54)*x)*P(n-1,x) - (267*n-735+(234*n-693)*x+(27*n-108)*x^2)*d/dx P(n-1,x) + (356+579*x+270*x^2+27*x^3)*d^2/dx^2 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[((45*n^2 - 153*n + 153 - (27*n - 54)*x)*P[n - 1, x] - (267*n - 735 + (234*n - 693)*x + (27*n - 108)*x^2)*D[P[n - 1, x], x] + (356 + 579*x + 270*x^2 + 27*x^3)*D[P[n - 1, x], {x, 2}])/(9*n)]];
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] = ((45*n^2 - 153*n + 153 - (27*n - 54)*x)*v[i - 1] - (267*n - 735 + (234*n - 693)*x + (27*n - 108)*x^2)*deriv(v[i - 1], x) + (356 + 579*x + 270*x^2 + 27*x^3)*deriv(deriv(v[i - 1], x), x))/(9*n)); 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

Candidate equalities ?

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