Real-rooted atlas

A257620 → D

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-18\,n + 36\,n x - 18\,n x^{2} - 18\,n^{2} x + 27\,n^{2} x^{2} - 9\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-9\,n x + 18\,n x^{2} - 9\,n x^{3} - 9\,n^{2} x^{2} + 9\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-6 + 6\,x + 9\,n - 12\,n x + 6\,n^{2} x\right)\,P_{n-1}(x) + \left(-3\,x + 3\,x^{2} + 3\,n x - 3\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
133
2361854
329759481972
4210610692631832419440
513851146772220158554041215437400
687480173647852837923472956437400437411022480
75401891877758210247625913663501246943955324113415309308629440

All 15 rows

Technical details
Node id
n-a06ada19a57d4cd7 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-b00b790aa67df404

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:       36,         18;
  3:      297,        594,          81;
  4:     2106,      10692,        6318,          324;
  5:    13851,     146772,      220158,        55404,         1215;
  6:    87480,    1736478,     5283792,      3472956,       437400,        4374;
  7:   540189,   18777582,   102476259,    136635012,     46943955,     3241134,      15309;
  8:  3293622,  191686176,  1736709822,   4099050360,   2894516370,   575058528,   23055354,     52488;
  9: 19938879, 1883269440, 26878632408, 103166791128, 128958488910, 53757264816, 6591443040, 159511032, 177147;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(18*n+(18*n^2-36*n)*x+(9*n^3-27*n^2+18*n)*x^2)*P(n-2,x) - (9*n*x+(9*n^2-18*n)*x^2-(9*n^2-9*n)*x^3)*d/dx P(n-2,x) + (9*n-6+(6*n^2-12*n+6)*x)*P(n-1,x) + ((3*n-3)*x-(3*n-3)*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[(-(18*n + (18*n^2 - 36*n)*x + (9*n^3 - 27*n^2 + 18*n)*x^2)*P[n - 2, x] - (9*n*x + (9*n^2 - 18*n)*x^2 - (9*n^2 - 9*n)*x^3)*D[P[n - 2, x], x] + (9*n - 6 + (6*n^2 - 12*n + 6)*x)*P[n - 1, x] + ((3*n - 3)*x - (3*n - 3)*x^2)*D[P[n - 1, x], x])/(n - 1)]];
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] = (-(18*n + (18*n^2 - 36*n)*x + (9*n^3 - 27*n^2 + 18*n)*x^2)*v[i - 2] - (9*n*x + (9*n^2 - 18*n)*x^2 - (9*n^2 - 9*n)*x^3)*deriv(v[i - 2], x) + (9*n - 6 + (6*n^2 - 12*n + 6)*x)*v[i - 1] + ((3*n - 3)*x - (3*n - 3)*x^2)*deriv(v[i - 1], x))/(n - 1)); 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. [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.