Real-rooted atlas

A046716 → basis A013610 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(32 + 48\,x + 18\,x^{2} - 16\,n - 24\,n x - 9\,n x^{2}\right)\,P_{n-2}(x) + \left(-3 - 3\,x + 4\,n + 3\,n x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 5 + 3\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2538
32933971
421735418027778
522974950371710808112125
6343499491798991466838829243284012
76749652319567318855621937237576741066777299241891
81627648167383420116153001106715853551252791518920117496002187378595022

All 15 rows

Technical details
Node id
n-9a60224d294916c0 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-40f5d787e72f16e0

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:           1;
   2:           5,           3;
   3:          29,          33,            9;
   4:         217,         354,          180,           27;
   5:        2297,        4950,         3717,         1080,           81;
   6:       34349,       94917,        98991,        46683,         8829,          243;
   7:      674965,     2319567,      3188556,      2193723,       757674,       106677,         729;
   8:    16276481,    67383420,    116153001,    106715853,     55125279,     15189201,     1749600,       2187;
   9:   461527793,  2241102780,   4660922790,   5381788068,   3726030699,   1546808724,   356540778,   35219448,      6561;
  10: 14993138773, 83631891159, 203986810890, 284158462230, 247262918139, 137622938949, 47846548854, 9499894974, 824776749, 19683;
%F Formula
The row polynomials satisfy P(n,x) = -(16*n-32+(24*n-48)*x+(9*n-18)*x^2)*P(n-2,x) + (4*n-3+(3*n-3)*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 5 + 3*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 5 + 3*x;
P[n_, x_] := P[n, x] = Expand[-(16*n - 32 + (24*n - 48)*x + (9*n - 18)*x^2)*P[n - 2, x] + (4*n - 3 + (3*n - 3)*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 5 + 3*x; for(n = 3, N, my(i = n); v[i] = -(16*n - 32 + (24*n - 48)*x + (9*n - 18)*x^2)*v[i - 2] + (4*n - 3 + (3*n - 3)*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

References

  1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [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
  3. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023