Real-rooted atlas

A081577 → basis A013611 → D

nonnegative coefficients

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

Recurrence

\(\left(-6 + 16\,x^{2} + 6\,n - 12\,n x - 16\,n x^{2} + 12\,n^{2} x\right)\,P_{n}(x) = \left(-48 - 264\,x - 352\,x^{2} - 128\,x^{3} + 78\,n + 348\,n x + 224\,n x^{2} - 18\,n^{2} - 60\,n^{2} x\right)\,P_{n-1}(x) + \left(-9 - 84\,x - 252\,x^{2} - 256\,x^{3} - 64\,x^{4} + 9\,n + 72\,n x + 156\,n x^{2} + 48\,n x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 4\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
144
2243256
396320192608
43521856268810245920
5120087042073618432512054192
639363635212134417817611264024576477024
712544140800606720126156813107206389761146884086016

All 15 rows

Technical details
Node id
n-6e8deaebfd093f63 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-253d2dd2c1570db3

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:      4;
  2:     24,      32;
  3:     96,     320,      192;
  4:    352,    1856,     2688,     1024;
  5:   1200,    8704,    20736,    18432,     5120;
  6:   3936,   36352,   121344,   178176,   112640,    24576;
  7:  12544,  140800,   606720,  1261568,  1310720,   638976,   114688;
  8:  39168,  517632,  2737152,  7409664, 10936320,  8699904,  3440640,   524288;
  9: 120384, 1830912, 11483136, 38412288, 74158080, 83755008, 53673984, 17825792, 2359296;
%F Formula
The row polynomials satisfy (6*n-6+(12*n^2-12*n)*x-(16*n-16)*x^2)*P(n,x) = -(18*n^2-78*n+48+(60*n^2-348*n+264)*x-(224*n-352)*x^2+128*x^3)*P(n-1,x) + (9*n-9+(72*n-84)*x+(156*n-252)*x^2+(48*n-256)*x^3-64*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[(-(18*n^2 - 78*n + 48 + (60*n^2 - 348*n + 264)*x - (224*n - 352)*x^2 + 128*x^3)*P[n - 1, x] + (9*n - 9 + (72*n - 84)*x + (156*n - 252)*x^2 + (48*n - 256)*x^3 - 64*x^4)*D[P[n - 1, x], x])/(6*n - 6 + (12*n^2 - 12*n)*x - (16*n - 16)*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] = 4; for(n = 2, N, my(i = n + 1); v[i] = (-(18*n^2 - 78*n + 48 + (60*n^2 - 348*n + 264)*x - (224*n - 352)*x^2 + 128*x^3)*v[i - 1] + (9*n - 9 + (72*n - 84)*x + (156*n - 252)*x^2 + (48*n - 256)*x^3 - 64*x^4)*deriv(v[i - 1], x))/(6*n - 6 + (12*n^2 - 12*n)*x - (16*n - 16)*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.