Real-rooted atlas

A081579 → basis A013610 → reverse

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(9\,n\right)\,P_{n}(x) = \left(90\,x^{2} + 40\,x^{3} + 27\,n - 36\,n x - 150\,n x^{2} - 60\,n x^{3} + 54\,n^{2} x + 60\,n^{2} x^{2} + 20\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(279\,x^{2} + 330\,x^{3} + 80\,x^{4} - 27\,n x - 198\,n x^{2} - 180\,n x^{3} - 40\,n x^{4}\right)\,P_{n-1}'(x) + \left(27\,x^{2} + 189\,x^{3} + 120\,x^{4} + 20\,x^{5}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1325
2924841
32712610824285
481540900480802081
5243210654005040192025614965
672977762721638880252007488832108121
7218727702123444244296235440116064282242688780045

All 15 rows

Technical details
Node id
n-90369a299eaa13f6 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-cb133f75685c9e6e

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:     3,      2;
  2:     9,     24,       8;
  3:    27,    126,     108,      24;
  4:    81,    540,     900,     480,       80;
  5:   243,   2106,    5400,    5040,     1920,      256;
  6:   729,   7776,   27216,   38880,    25200,     7488,     832;
  7:  2187,  27702,  123444,  244296,   235440,   116064,   28224,    2688;
  8:  6561,  96228,  521964, 1337472,  1770336,  1276992,  507456,  104448,   8704;
  9: 19683, 328050, 2099520, 6648480, 11411280, 11174112, 6410880, 2131200, 380160, 28160;
%F Formula
The row polynomials satisfy 9*n*P(n,x) = (27*n+(54*n^2-36*n)*x+(60*n^2-150*n+90)*x^2+(20*n^2-60*n+40)*x^3)*P(n-1,x) - (27*n*x+(198*n-279)*x^2+(180*n-330)*x^3+(40*n-80)*x^4)*d/dx P(n-1,x) + (27*x^2+189*x^3+120*x^4+20*x^5)*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 + (54*n^2 - 36*n)*x + (60*n^2 - 150*n + 90)*x^2 + (20*n^2 - 60*n + 40)*x^3)*P[n - 1, x] - (27*n*x + (198*n - 279)*x^2 + (180*n - 330)*x^3 + (40*n - 80)*x^4)*D[P[n - 1, x], x] + (27*x^2 + 189*x^3 + 120*x^4 + 20*x^5)*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 + (54*n^2 - 36*n)*x + (60*n^2 - 150*n + 90)*x^2 + (20*n^2 - 60*n + 40)*x^3)*v[i - 1] - (27*n*x + (198*n - 279)*x^2 + (180*n - 330)*x^3 + (40*n - 80)*x^4)*deriv(v[i - 1], x) + (27*x^2 + 189*x^3 + 120*x^4 + 20*x^5)*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

Same fingerprint ?

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