Real-rooted atlas

A116412 → basis A013610 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(9 + 9\,n\right)\,P_{n}(x) = \left(252 + 108\,x - 153\,n - 54\,n x + 27\,n^{2}\right)\,P_{n-1}(x) + \left(600 + 612\,x + 135\,x^{2} - 156\,n - 144\,n x - 27\,n x^{2}\right)\,P_{n-1}'(x) + \left(208 + 348\,x + 180\,x^{2} + 27\,x^{3}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
17310
24642997
330442318927943
4200837562565756819166
513264311642868312906283524389095
6876162478242871181742045832010206729866017
75787521914240267440420442519225902456733572121878417818

All 15 rows

Technical details
Node id
n-9ec3236948b44c1d · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-da8182181d1a9cd6

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:        7,         3;
  2:       46,        42,         9;
  3:      304,       423,       189,        27;
  4:     2008,      3756,      2565,       756,        81;
  5:    13264,     31164,     28683,     12906,      2835,      243;
  6:    87616,    247824,    287118,    174204,     58320,    10206,      729;
  7:   578752,   1914240,   2674404,   2044251,    922590,   245673,    35721,    2187;
  8:  3822976,  14475840,  23681088,  21846888,  12424185,  4457592,   984879,  122472,   6561;
  9: 25252864, 107717184, 201954384, 218324808, 149908887, 67768569, 20161953, 3805380, 413343, 19683;
%F Formula
The row polynomials satisfy (9*n+9)*P(n,x) = (27*n^2-153*n+252-(54*n-108)*x)*P(n-1,x) - (156*n-600+(144*n-612)*x+(27*n-135)*x^2)*d/dx P(n-1,x) + (208+348*x+180*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[((27*n^2 - 153*n + 252 - (54*n - 108)*x)*P[n - 1, x] - (156*n - 600 + (144*n - 612)*x + (27*n - 135)*x^2)*D[P[n - 1, x], x] + (208 + 348*x + 180*x^2 + 27*x^3)*D[P[n - 1, x], {x, 2}])/(9*n + 9)]];
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] = ((27*n^2 - 153*n + 252 - (54*n - 108)*x)*v[i - 1] - (156*n - 600 + (144*n - 612)*x + (27*n - 135)*x^2)*deriv(v[i - 1], x) + (208 + 348*x + 180*x^2 + 27*x^3)*deriv(deriv(v[i - 1], x), x))/(9*n + 9)); 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