Real-rooted atlas

A142461 → D

nonnegative coefficients

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

Recurrence

\(\left(-11 + 6\,n\right)\,P_{n}(x) = \left(35 - 170\,x + 935\,x^{2} - 42\,n + 264\,n x - 1962\,n x^{2} - 72\,n^{2} x + 1188\,n^{2} x^{2} - 216\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(30\,x - 360\,x^{2} + 330\,x^{3} - 36\,n x + 612\,n x^{2} - 576\,n x^{3} - 216\,n^{2} x^{2} + 216\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-82 + 242\,x + 48\,n - 264\,n x + 72\,n^{2} x\right)\,P_{n-1}(x) + \left(-66\,x + 66\,x^{2} + 36\,n x - 36\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
111
214216
31112223336
47965772238848960
55597105284157926223885291200
639210164854260011883297084196050611182080
7274507238620661758428972344571965965516516470427495738880

All 15 rows

Technical details
Node id
n-743f2f9ce8f29f74 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-8db21305723b142a

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:        1;
  2:       14,          2;
  3:      111,        222,            3;
  4:      796,       5772,         2388,            4;
  5:     5597,     105284,       157926,        22388,            5;
  6:    39210,    1648542,      6001188,      3297084,       196050,            6;
  7:   274507,   23862066,    175842897,    234457196,     59655165,      1647042,           7;
  8:  1921592,  330520376,   4450601112,  11722859800,   7417668520,    991561128,    13451144,         8;
  9: 13451193, 4462021800, 102905301996, 477029674296, 596287092870, 205810603992, 15617076300, 107609544, 9;
%F Formula
The row polynomials satisfy (6*n-11)*P(n,x) = -(42*n-35+(72*n^2-264*n+170)*x+(216*n^3-1188*n^2+1962*n-935)*x^2)*P(n-2,x) - ((36*n-30)*x+(216*n^2-612*n+360)*x^2-(216*n^2-576*n+330)*x^3)*d/dx P(n-2,x) + (48*n-82+(72*n^2-264*n+242)*x)*P(n-1,x) + ((36*n-66)*x-(36*n-66)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(42*n - 35 + (72*n^2 - 264*n + 170)*x + (216*n^3 - 1188*n^2 + 1962*n - 935)*x^2)*P[n - 2, x] - ((36*n - 30)*x + (216*n^2 - 612*n + 360)*x^2 - (216*n^2 - 576*n + 330)*x^3)*D[P[n - 2, x], x] + (48*n - 82 + (72*n^2 - 264*n + 242)*x)*P[n - 1, x] + ((36*n - 66)*x - (36*n - 66)*x^2)*D[P[n - 1, x], x])/(6*n - 11)]];
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] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(42*n - 35 + (72*n^2 - 264*n + 170)*x + (216*n^3 - 1188*n^2 + 1962*n - 935)*x^2)*v[i - 2] - ((36*n - 30)*x + (216*n^2 - 612*n + 360)*x^2 - (216*n^2 - 576*n + 330)*x^3)*deriv(v[i - 2], x) + (48*n - 82 + (72*n^2 - 264*n + 242)*x)*v[i - 1] + ((36*n - 66)*x - (36*n - 66)*x^2)*deriv(v[i - 1], x))/(6*n - 11)); 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.