Real-rooted atlas

A257609 → D

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-8\,n + 16\,n x - 8\,n x^{2} - 8\,n^{2} x + 12\,n^{2} x^{2} - 4\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-4\,n x + 8\,n x^{2} - 4\,n x^{3} - 4\,n^{2} x^{2} + 4\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-4 + 4\,x + 6\,n - 8\,n x + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(-2\,x + 2\,x^{2} + 2\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
122
216824
38817624288
441621121248643840
518241932828992729616057600
6768015244846387230489638400384967680
731616109900859976967996928274752018969689618063360

All 15 rows

Technical details
Node id
n-627ac699506485c3 · 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:      2;
  2:     16,        8;
  3:     88,      176,        24;
  4:    416,     2112,      1248,         64;
  5:   1824,    19328,     28992,       7296,        160;
  6:   7680,   152448,    463872,     304896,      38400,        384;
  7:  31616,  1099008,   5997696,    7996928,    2747520,     189696,       896;
  8: 128512,  7479296,  67763712,  159938560,  112939520,   22437888,    899584,    2048;
  9: 518656, 48988160, 699174912, 2683604992, 3354506240, 1398349824, 171458560, 4149248, 4608;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(8*n+(8*n^2-16*n)*x+(4*n^3-12*n^2+8*n)*x^2)*P(n-2,x) - (4*n*x+(4*n^2-8*n)*x^2-(4*n^2-4*n)*x^3)*d/dx P(n-2,x) + (6*n-4+(4*n^2-8*n+4)*x)*P(n-1,x) + ((2*n-2)*x-(2*n-2)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[(-(8*n + (8*n^2 - 16*n)*x + (4*n^3 - 12*n^2 + 8*n)*x^2)*P[n - 2, x] - (4*n*x + (4*n^2 - 8*n)*x^2 - (4*n^2 - 4*n)*x^3)*D[P[n - 2, x], x] + (6*n - 4 + (4*n^2 - 8*n + 4)*x)*P[n - 1, x] + ((2*n - 2)*x - (2*n - 2)*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] = 2; for(n = 2, N, my(i = n + 1); v[i] = (-(8*n + (8*n^2 - 16*n)*x + (4*n^3 - 12*n^2 + 8*n)*x^2)*v[i - 2] - (4*n*x + (4*n^2 - 8*n)*x^2 - (4*n^2 - 4*n)*x^3)*deriv(v[i - 2], x) + (6*n - 4 + (4*n^2 - 8*n + 4)*x)*v[i - 1] + ((2*n - 2)*x - (2*n - 2)*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.