Real-rooted atlas

A126216 → basis A013610 → D

nonnegative coefficients

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

Recurrence

\(\left(-6 + 3\,n + 3\,n^{2}\right)\,P_{n}(x) = \left(-6 - 18\,x + 3\,n - 9\,n x + 21\,n^{2} + 27\,n^{2} x\right)\,P_{n-1}(x) + \left(-2 - 9\,x - 9\,x^{2} - 4\,n - 18\,n x - 18\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
133
2211839
312621681423
4726187214583244380
5412514220170108100121544670
623319100710163215121500400954374453213
713162868342414004901428840748440183708153094591839

All 15 rows

Technical details
Node id
n-41b301a93d32aa8a · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-3abc2859e25348b7

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:       0;
  1:       3;
  2:      21,       18;
  3:     126,      216,       81;
  4:     726,     1872,     1458,       324;
  5:    4125,    14220,    17010,      8100,      1215;
  6:   23319,   100710,   163215,    121500,     40095,     4374;
  7:  131628,   683424,  1400490,   1428840,    748440,   183708,    15309;
  8:  743052,  4507776, 11181240,  14542416,  10546200,  4164048,   796068,   52488;
  9: 4197879, 29141208, 84900312, 134474256, 125227620, 69074208, 21493836, 3306744, 177147;
%F Formula
The row polynomials satisfy (3*n^2+3*n-6)*P(n,x) = (21*n^2+3*n-6+(27*n^2-9*n-18)*x)*P(n-1,x) - (4*n+2+(18*n+9)*x+(18*n+9)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[((21*n^2 + 3*n - 6 + (27*n^2 - 9*n - 18)*x)*P[n - 1, x] - (4*n + 2 + (18*n + 9)*x + (18*n + 9)*x^2)*D[P[n - 1, x], x])/(3*n^2 + 3*n - 6)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 3; for(n = 2, N, my(i = n + 1); v[i] = ((21*n^2 + 3*n - 6 + (27*n^2 - 9*n - 18)*x)*v[i - 1] - (4*n + 2 + (18*n + 9)*x + (18*n + 9)*x^2)*deriv(v[i - 1], x))/(3*n^2 + 3*n - 6)); 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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023