Real-rooted atlas

A111125 → basis A013611 → reverse

nonnegative coefficients

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

Recurrence

\(\left(-12 + 8\,n\right)\,P_{n}(x) = \left(-48 - 56\,x - 10\,x^{2} + 32\,n + 36\,n x + 5\,n x^{2}\right)\,P_{n-1}(x) + \left(-16\,x - 24\,x^{2} - 5\,x^{3}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2448
316281155
46416012429377
5256832960460762584
61024409662724560156019917711
74096194563712036224189605016521121393
8163849011220582425216017824072384155721364832040

All 15 rows

Technical details
Node id
n-95cc8af35ee1eba1 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-8805e6f0a07b591c

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:      4,       4;
   3:     16,      28,      11;
   4:     64,     160,     124,      29;
   5:    256,     832,     960,     460,       76;
   6:   1024,    4096,    6272,    4560,     1560,     199;
   7:   4096,   19456,   37120,   36224,    18960,    5016,     521;
   8:  16384,   90112,  205824,  252160,   178240,   72384,   15572,   1364;
   9:  65536,  409600, 1089536, 1606656,  1432320,  788032,  260480,  47156,   3571;
  10: 262144, 1835008, 5570560, 9605120, 10343424, 7196928, 3227776, 897680, 140180, 9349;
%F Formula
The row polynomials satisfy (8*n-12)*P(n,x) = (32*n-48+(36*n-56)*x+(5*n-10)*x^2)*P(n-1,x) - (16*x+24*x^2+5*x^3)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((32*n - 48 + (36*n - 56)*x + (5*n - 10)*x^2)*P[n - 1, x] - (16*x + 24*x^2 + 5*x^3)*D[P[n - 1, x], x])/(8*n - 12)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((32*n - 48 + (36*n - 56)*x + (5*n - 10)*x^2)*v[i - 1] - (16*x + 24*x^2 + 5*x^3)*deriv(v[i - 1], x))/(8*n - 12)); 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

Same fingerprint ?

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