Real-rooted atlas

A257611 → D

nonnegative coefficients

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

Recurrence

\(\left(-1 + 2\,n\right)\,P_{n}(x) = \left(-15 + 18\,x - 3\,x^{2} - 30\,n + 24\,n x + 2\,n x^{2} - 24\,n^{2} x + 12\,n^{2} x^{2} - 8\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-6\,x + 8\,x^{2} - 2\,x^{3} - 12\,n x + 12\,n x^{2} - 8\,n^{2} x^{2} + 8\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-2 + 2\,x + 16\,n - 8\,n x + 8\,n^{2} x\right)\,P_{n-1}(x) + \left(-2\,x + 2\,x^{2} + 4\,n x - 4\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
133
2301848
321342681720
413085964392432411520
574316529297938297241215201600
6403146205261762884124105220157043743870720
7212505539184626104635348061801347961512750301530981285120

All 15 rows

Technical details
Node id
n-ea6477fe3049a666 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-a1d3ab53fbcf8658

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:      30,        18;
  3:     213,       426,         81;
  4:    1308,      5964,       3924,         324;
  5:    7431,     65292,      97938,       29724,        1215;
  6:   40314,    620526,    1762884,     1241052,      201570,       4374;
  7:  212505,   5391846,   26104635,    34806180,    13479615,    1275030,      15309;
  8: 1099704,  44118072,  340082712,   765735960,   566804520,  132354216,    7697928,    52488;
  9: 5623179, 346216440, 4053401028, 14317862568, 17897328210, 8106802056, 1211757540, 44985432, 177147;
%F Formula
The row polynomials satisfy (2*n-1)*P(n,x) = -(30*n+15+(24*n^2-24*n-18)*x+(8*n^3-12*n^2-2*n+3)*x^2)*P(n-2,x) - ((12*n+6)*x+(8*n^2-12*n-8)*x^2-(8*n^2-2)*x^3)*d/dx P(n-2,x) + (16*n-2+(8*n^2-8*n+2)*x)*P(n-1,x) + ((4*n-2)*x-(4*n-2)*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[(-(30*n + 15 + (24*n^2 - 24*n - 18)*x + (8*n^3 - 12*n^2 - 2*n + 3)*x^2)*P[n - 2, x] - ((12*n + 6)*x + (8*n^2 - 12*n - 8)*x^2 - (8*n^2 - 2)*x^3)*D[P[n - 2, x], x] + (16*n - 2 + (8*n^2 - 8*n + 2)*x)*P[n - 1, x] + ((4*n - 2)*x - (4*n - 2)*x^2)*D[P[n - 1, x], x])/(2*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] = (-(30*n + 15 + (24*n^2 - 24*n - 18)*x + (8*n^3 - 12*n^2 - 2*n + 3)*x^2)*v[i - 2] - ((12*n + 6)*x + (8*n^2 - 12*n - 8)*x^2 - (8*n^2 - 2)*x^3)*deriv(v[i - 2], x) + (16*n - 2 + (8*n^2 - 8*n + 2)*x)*v[i - 1] + ((4*n - 2)*x - (4*n - 2)*x^2)*deriv(v[i - 1], x))/(2*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

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.