Real-rooted atlas

A121574 → basis A013610 → reverse

nonnegative coefficients

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

Recurrence

\(\left(9\,n\right)\,P_{n}(x) = \left(54\,x^{2} + 26\,x^{3} + 27\,n - 9\,n x - 84\,n x^{2} - 39\,n x^{3} + 36\,n^{2} x + 30\,n^{2} x^{2} + 13\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(144\,x^{2} + 192\,x^{3} + 52\,x^{4} - 27\,n x - 108\,n x^{2} - 99\,n x^{3} - 26\,n x^{4}\right)\,P_{n-1}'(x) + \left(27\,x^{2} + 99\,x^{3} + 69\,x^{4} + 13\,x^{5}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1336
29211040
3279910533264
4814056844651091744
5243153935913861192636011520
672955891668624813194407650118976096
72187196837168513680914793390909295233927502656

All 15 rows

Technical details
Node id
n-6622f652d66da1b7 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-16deb54cf5efeb3a

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,      3;
  2:     9,     21,      10;
  3:    27,     99,     105,      33;
  4:    81,    405,     684,     465,     109;
  5:   243,   1539,    3591,    3861,    1926,     360;
  6:   729,   5589,   16686,   24813,   19440,    7650,    1189;
  7:  2187,  19683,   71685,  136809,  147933,   90909,   29523,    3927;
  8:  6561,  67797,  291600,  681129,  945351,  799659,  403686,  111567,  12970;
  9: 19683, 229635, 1139427, 3152925, 5361552, 5815638, 4030695, 1725237, 414915, 42837;
%F Formula
The row polynomials satisfy 9*n*P(n,x) = (27*n+(36*n^2-9*n)*x+(30*n^2-84*n+54)*x^2+(13*n^2-39*n+26)*x^3)*P(n-1,x) - (27*n*x+(108*n-144)*x^2+(99*n-192)*x^3+(26*n-52)*x^4)*d/dx P(n-1,x) + (27*x^2+99*x^3+69*x^4+13*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 + (36*n^2 - 9*n)*x + (30*n^2 - 84*n + 54)*x^2 + (13*n^2 - 39*n + 26)*x^3)*P[n - 1, x] - (27*n*x + (108*n - 144)*x^2 + (99*n - 192)*x^3 + (26*n - 52)*x^4)*D[P[n - 1, x], x] + (27*x^2 + 99*x^3 + 69*x^4 + 13*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 + (36*n^2 - 9*n)*x + (30*n^2 - 84*n + 54)*x^2 + (13*n^2 - 39*n + 26)*x^3)*v[i - 1] - (27*n*x + (108*n - 144)*x^2 + (99*n - 192)*x^3 + (26*n - 52)*x^4)*deriv(v[i - 1], x) + (27*x^2 + 99*x^3 + 69*x^4 + 13*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