Real-rooted atlas

A232535 → basis A013610 → reverse

nonnegative coefficients

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

Recurrence

\(\left(n^{2}\right)\,P_{n}(x) = \left(-3\,x - 2\,x^{2} + 3\,n + 4\,n x + 4\,n x^{2} + 3\,n^{2} + 2\,n^{2} x - 2\,n^{2} x^{2}\right)\,P_{n-1}(x) + \left(3\,x^{2} + 2\,x^{3} + 6\,n x + 2\,n x^{2}\right)\,P_{n-1}'(x) + \left(6\,x^{3} + 2\,x^{4}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1639
227421281
310833324048729
44052052266412481926561
51458109352203217712614476859049
6510352974151956183168105984291843072531441
71749623984192534415474241313280592128135168122884782969

All 15 rows

Technical details
Node id
n-8f401ed332c8108e · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-eec54e4acac10975

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:     27,      42,       12;
  3:    108,     333,      240,       48;
  4:    405,    2052,     2664,     1248,       192;
  5:   1458,   10935,    22032,    17712,      6144,      768;
  6:   5103,   52974,   151956,   183168,    105984,    29184,     3072;
  7:  17496,  239841,   925344,  1547424,   1313280,   592128,   135168,    12288;
  8:  59049, 1032264,  5143824, 11337408,  13115520,  8543232,  3151872,   614400,   49152;
  9: 196830, 4271211, 26663904, 74672928, 112223232, 98392320, 51867648, 16183296, 2752512, 196608;
%F Formula
The row polynomials satisfy n^2*P(n,x) = (3*n^2+3*n+(2*n^2+4*n-3)*x-(2*n^2-4*n+2)*x^2)*P(n-1,x) + (6*n*x+(2*n+3)*x^2+2*x^3)*d/dx P(n-1,x) + (6*x^3+2*x^4)*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[((3*n^2 + 3*n + (2*n^2 + 4*n - 3)*x - (2*n^2 - 4*n + 2)*x^2)*P[n - 1, x] + (6*n*x + (2*n + 3)*x^2 + 2*x^3)*D[P[n - 1, x], x] + (6*x^3 + 2*x^4)*D[P[n - 1, x], {x, 2}])/(n^2)]];
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] = ((3*n^2 + 3*n + (2*n^2 + 4*n - 3)*x - (2*n^2 - 4*n + 2)*x^2)*v[i - 1] + (6*n*x + (2*n + 3)*x^2 + 2*x^3)*deriv(v[i - 1], x) + (6*x^3 + 2*x^4)*deriv(deriv(v[i - 1], x), x))/(n^2)); 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. [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