Real-rooted atlas

A113214 → basis A013610 → D

nonnegative coefficients

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

Recurrence

\(\left(9\,n + 9\,n x - 12\,n^{2} - 9\,n^{2} x + 3\,n^{3}\right)\,P_{n}(x) = \left(60 + 36\,x - 135\,n - 171\,n x - 54\,n x^{2} + 57\,n^{2} + 45\,n^{2} x - 6\,n^{3}\right)\,P_{n-1}(x) + \left(10 + 36\,x + 18\,x^{2} - 20\,n - 87\,n x - 90\,n x^{2} - 27\,n x^{3} + 5\,n^{2} + 18\,n^{2} x + 9\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
133
2151833
33912681246
4874687293241608
518013503483356412159792
63543420123932073615795437457072
7675797437341878041081356561015309322848

All 15 rows

Technical details
Node id
n-d8a420e53a63605e · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-c0c03255b05775de

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:   15,    18;
  3:   39,   126,     81;
  4:   87,   468,    729,    324;
  5:  180,  1350,   3483,   3564,    1215;
  6:  354,  3420,  12393,  20736,   15795,    4374;
  7:  675,  7974,  37341,  87804,  108135,   65610,   15309;
  8: 1257, 17568, 101007, 307476,  530955,  516132,  260253,  52488;
  9: 2301, 37134, 253287, 948672, 2121390, 2882466, 2311659, 997272, 177147;
%F Formula
The row polynomials satisfy (3*n^3-12*n^2+9*n-(9*n^2-9*n)*x)*P(n,x) = -(6*n^3-57*n^2+135*n-60-(45*n^2-171*n+36)*x+54*n*x^2)*P(n-1,x) + (5*n^2-20*n+10+(18*n^2-87*n+36)*x+(9*n^2-90*n+18)*x^2-27*n*x^3)*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[(-(6*n^3 - 57*n^2 + 135*n - 60 - (45*n^2 - 171*n + 36)*x + 54*n*x^2)*P[n - 1, x] + (5*n^2 - 20*n + 10 + (18*n^2 - 87*n + 36)*x + (9*n^2 - 90*n + 18)*x^2 - 27*n*x^3)*D[P[n - 1, x], x])/(3*n^3 - 12*n^2 + 9*n - (9*n^2 - 9*n)*x)]];
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] = (-(6*n^3 - 57*n^2 + 135*n - 60 - (45*n^2 - 171*n + 36)*x + 54*n*x^2)*v[i - 1] + (5*n^2 - 20*n + 10 + (18*n^2 - 87*n + 36)*x + (9*n^2 - 90*n + 18)*x^2 - 27*n*x^3)*deriv(v[i - 1], x))/(3*n^3 - 12*n^2 + 9*n - (9*n^2 - 9*n)*x)); 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

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
  2. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.