Real-rooted atlas

A026729 → basis A013610 → D

nonnegative coefficients

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

Recurrence

\(\left(12 + 18\,x - 12\,n - 9\,n x + 3\,n^{2}\right)\,P_{n}(x) = \left(-90 - 144\,x - 54\,x^{2} + 51\,n + 45\,n x - 6\,n^{2}\right)\,P_{n-1}(x) + \left(-15 - 69\,x - 81\,x^{2} - 27\,x^{3} + 5\,n + 18\,n x + 9\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
10
233
391827
4219081192
5452885673241224
6907742349291612157344
7174187276951490413365437442384
832742302195157996814055686215309238080

All 15 rows

Technical details
Node id
n-6b70b892818f1315 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-be24edfd5cc5a132

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 -2)
  • 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:    0;
   2:    3;
   3:    9,    18;
   4:   21,    90,     81;
   5:   45,   288,    567,    324;
   6:   90,   774,   2349,   2916,    1215;
   7:  174,  1872,   7695,  14904,   13365,    4374;
   8:  327,  4230,  21951,  57996,   81405,   56862,   15309;
   9:  603,  9108,  57105, 191484,  368145,  402408,  229635,  52488;
  10: 1095, 18918, 139077, 565704, 1385100, 2077650, 1852389, 892296, 177147;
%F Formula
The row polynomials satisfy (3*n^2-12*n+12-(9*n-18)*x)*P(n,x) = -(6*n^2-51*n+90-(45*n-144)*x+54*x^2)*P(n-1,x) + (5*n-15+(18*n-69)*x+(9*n-81)*x^2-27*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 3.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[(-(6*n^2 - 51*n + 90 - (45*n - 144)*x + 54*x^2)*P[n - 1, x] + (5*n - 15 + (18*n - 69)*x + (9*n - 81)*x^2 - 27*x^3)*D[P[n - 1, x], x])/(3*n^2 - 12*n + 12 - (9*n - 18)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 3; for(n = 3, N, my(i = n); v[i] = (-(6*n^2 - 51*n + 90 - (45*n - 144)*x + 54*x^2)*v[i - 1] + (5*n - 15 + (18*n - 69)*x + (9*n - 81)*x^2 - 27*x^3)*deriv(v[i - 1], x))/(3*n^2 - 12*n + 12 - (9*n - 18)*x)); 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. [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.