Real-rooted atlas

A121574 → basis A013610 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(9\,n\right)\,P_{n}(x) = \left(108 + 54\,x - 81\,n - 27\,n x + 27\,n^{2}\right)\,P_{n-1}(x) + \left(444 + 468\,x + 108\,x^{2} - 156\,n - 144\,n x - 27\,n x^{2}\right)\,P_{n-1}'(x) + \left(208 + 348\,x + 180\,x^{2} + 27\,x^{3}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1639
24039988
326438418027855
4174433722385729818311
511520277922629812177275424380784
676096220032260820162027555669963729785233
75026561694208241358418822248670242357103499221877632585

All 15 rows

Technical details
Node id
n-abd693d5e453ae4d · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-ad215bfd092541fc

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:        6,        3;
  2:       40,       39,         9;
  3:      264,      384,       180,        27;
  4:     1744,     3372,      2385,       729,        81;
  5:    11520,    27792,     26298,     12177,      2754,      243;
  6:    76096,   220032,    260820,    162027,     55566,     9963,      729;
  7:   502656,  1694208,   2413584,   1882224,    867024,   235710,    34992,    2187;
  8:  3320320, 12781632,  21267504,  19964664,  11557161,  4221882,   949887,  120285,   6561;
  9: 21932544, 94935552, 180686880, 198360144, 138351726, 63546687, 19212066, 3685095, 406782, 19683;
%F Formula
The row polynomials satisfy 9*n*P(n,x) = (27*n^2-81*n+108-(27*n-54)*x)*P(n-1,x) - (156*n-444+(144*n-468)*x+(27*n-108)*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 - 81*n + 108 - (27*n - 54)*x)*P[n - 1, x] - (156*n - 444 + (144*n - 468)*x + (27*n - 108)*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)]];
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 - 81*n + 108 - (27*n - 54)*x)*v[i - 1] - (156*n - 444 + (144*n - 468)*x + (27*n - 108)*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)); 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