Real-rooted atlas

A142458 → D

nonnegative coefficients

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

Recurrence

\(\left(-5 + 3\,n\right)\,P_{n}(x) = \left(8 - 32\,x + 80\,x^{2} - 12\,n + 60\,n x - 198\,n x^{2} - 18\,n^{2} x + 135\,n^{2} x^{2} - 27\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(6\,x - 36\,x^{2} + 30\,x^{3} - 9\,n x + 72\,n x^{2} - 63\,n x^{3} - 27\,n^{2} x^{2} + 27\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-22 + 50\,x + 15\,n - 60\,n x + 18\,n^{2} x\right)\,P_{n-1}(x) + \left(-15\,x + 15\,x^{2} + 9\,n x - 9\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
111
28210
339783120
4166109249841760
567710964164462708530800
6272494350328920188700136206628320
71091574761851290256838700186904565490714660800

All 15 rows

Technical details
Node id
n-f1df042cd3a7a7cc · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-d87433d07b949f42

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:      8,        2;
  3:     39,       78,         3;
  4:    166,     1092,       498,          4;
  5:    677,    10964,     16446,       2708,          5;
  6:   2724,    94350,    328920,     188700,      13620,          6;
  7:  10915,   747618,   5129025,    6838700,    1869045,      65490,         7;
  8:  43682,  5648096,  69233082,  177806200,  115388470,   16944288,    305774,       8;
  9: 174753, 41458680, 853301556, 3788453016, 4735566270, 1706603112, 145105380, 1398024, 9;
%F Formula
The row polynomials satisfy (3*n-5)*P(n,x) = -(12*n-8+(18*n^2-60*n+32)*x+(27*n^3-135*n^2+198*n-80)*x^2)*P(n-2,x) - ((9*n-6)*x+(27*n^2-72*n+36)*x^2-(27*n^2-63*n+30)*x^3)*d/dx P(n-2,x) + (15*n-22+(18*n^2-60*n+50)*x)*P(n-1,x) + ((9*n-15)*x-(9*n-15)*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[(-(12*n - 8 + (18*n^2 - 60*n + 32)*x + (27*n^3 - 135*n^2 + 198*n - 80)*x^2)*P[n - 2, x] - ((9*n - 6)*x + (27*n^2 - 72*n + 36)*x^2 - (27*n^2 - 63*n + 30)*x^3)*D[P[n - 2, x], x] + (15*n - 22 + (18*n^2 - 60*n + 50)*x)*P[n - 1, x] + ((9*n - 15)*x - (9*n - 15)*x^2)*D[P[n - 1, x], x])/(3*n - 5)]];
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] = (-(12*n - 8 + (18*n^2 - 60*n + 32)*x + (27*n^3 - 135*n^2 + 198*n - 80)*x^2)*v[i - 2] - ((9*n - 6)*x + (27*n^2 - 72*n + 36)*x^2 - (27*n^2 - 63*n + 30)*x^3)*deriv(v[i - 2], x) + (15*n - 22 + (18*n^2 - 60*n + 50)*x)*v[i - 1] + ((9*n - 15)*x - (9*n - 15)*x^2)*deriv(v[i - 1], x))/(3*n - 5)); 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.