Real-rooted atlas

A081578 → basis A013610 → D

nonnegative coefficients

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

Recurrence

\(\left(-24 + 27\,x^{2} + 24\,n - 36\,n x - 27\,n x^{2} + 36\,n^{2} x\right)\,P_{n}(x) = \left(-264 - 1008\,x - 783\,x^{2} - 162\,x^{3} + 408\,n + 1332\,n x + 513\,n x^{2} - 96\,n^{2} - 252\,n^{2} x\right)\,P_{n-1}(x) + \left(-64 - 432\,x - 900\,x^{2} - 567\,x^{3} - 81\,x^{4} + 64\,n + 384\,n x + 612\,n x^{2} + 108\,n x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
133
2211839
39021681387
4366140413773243471
513657272122317128121529211
64923335348027180676328054374236583
717220142992446310650592451980139968153091864371

All 15 rows

Technical details
Node id
n-da51837902eee65b · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-c22c71caf2d89bf9

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:     90,     216,       81;
  4:    366,    1404,     1377,      324;
  5:   1365,    7272,    12231,     7128,     1215;
  6:   4923,   33534,    80271,    80676,    32805,     4374;
  7:  17220,  142992,   446310,   650592,   451980,   139968,    15309;
  8:  59052,  578520,  2229606,  4291704,  4349700,  2274480,   566433,   52488;
  9: 199287, 2250288, 10331874, 24809328, 33468390, 25509168, 10609137, 2204496, 177147;
%F Formula
The row polynomials satisfy (24*n-24+(36*n^2-36*n)*x-(27*n-27)*x^2)*P(n,x) = -(96*n^2-408*n+264+(252*n^2-1332*n+1008)*x-(513*n-783)*x^2+162*x^3)*P(n-1,x) + (64*n-64+(384*n-432)*x+(612*n-900)*x^2+(108*n-567)*x^3-81*x^4)*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[(-(96*n^2 - 408*n + 264 + (252*n^2 - 1332*n + 1008)*x - (513*n - 783)*x^2 + 162*x^3)*P[n - 1, x] + (64*n - 64 + (384*n - 432)*x + (612*n - 900)*x^2 + (108*n - 567)*x^3 - 81*x^4)*D[P[n - 1, x], x])/(24*n - 24 + (36*n^2 - 36*n)*x - (27*n - 27)*x^2)]];
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] = (-(96*n^2 - 408*n + 264 + (252*n^2 - 1332*n + 1008)*x - (513*n - 783)*x^2 + 162*x^3)*v[i - 1] + (64*n - 64 + (384*n - 432)*x + (612*n - 900)*x^2 + (108*n - 567)*x^3 - 81*x^4)*deriv(v[i - 1], x))/(24*n - 24 + (36*n^2 - 36*n)*x - (27*n - 27)*x^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

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.