Real-rooted atlas

A060187 → γ → D

nonnegative coefficients

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

Recurrence

\(\left(-3 + n\right)\,P_{n}(x) = \left(6 - 80\,x + 480\,x^{2} - 3\,n + 56\,n x - 496\,n x^{2} - 8\,n^{2} x + 160\,n^{2} x^{2} - 16\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(4\,x - 64\,x^{2} + 192\,x^{3} - 2\,n x + 48\,n x^{2} - 160\,n x^{3} - 8\,n^{2} x^{2} + 32\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-11 + 96\,x + 4\,n - 56\,n x + 8\,n^{2} x\right)\,P_{n-1}(x) + \left(-6\,x + 24\,x^{2} + 2\,n x - 8\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 0\)
\(P_{3}(x) = 4\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
10
20
344
42020
572160232
623219522184
7716153281171227756
8217299552265920367644
96544584640365337614182405662800
10196643237312396049925173350494595472
11590281728774437435430410992814082586675201749650004
12177124902165123242838912178797578241383815680035051147172
135314164638354882648896588824739352473642041609216066423152640761186102328
141594296236169072020768686464030741886310409572447508480489949546905617756181758232
1547829401194875164815813393513603542134062592018215163836928020153668521984022864312188928443567269289916
Technical details
Node id
n-18bae3f267a0e827 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-f9921f80f8a8f61e

Observed on the stored rows ?

  • Degrees: -1, -1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6 (half affine: slope 1, intercept -3)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:     0;
   2:     0;
   3:     4;
   4:    20;
   5:    72,     160;
   6:   232,    1952;
   7:   716,   15328,    11712;
   8:  2172,   99552,   265920;
   9:  6544,  584640,  3653376,  1418240;
  10: 19664, 3237312, 39604992, 51733504;
%F Formula
The row polynomials satisfy (n-3)*P(n,x) = -(3*n-6+(8*n^2-56*n+80)*x+(16*n^3-160*n^2+496*n-480)*x^2)*P(n-2,x) - ((2*n-4)*x+(8*n^2-48*n+64)*x^2-(32*n^2-160*n+192)*x^3)*d/dx P(n-2,x) + (4*n-11+(8*n^2-56*n+96)*x)*P(n-1,x) + ((2*n-6)*x-(8*n-24)*x^2)*d/dx P(n-1,x) for n >= 4, with P(1,x) = 0, P(2,x) = 0, P(3,x) = 4.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 0;
P[3, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[(-(3*n - 6 + (8*n^2 - 56*n + 80)*x + (16*n^3 - 160*n^2 + 496*n - 480)*x^2)*P[n - 2, x] - ((2*n - 4)*x + (8*n^2 - 48*n + 64)*x^2 - (32*n^2 - 160*n + 192)*x^3)*D[P[n - 2, x], x] + (4*n - 11 + (8*n^2 - 56*n + 96)*x)*P[n - 1, x] + ((2*n - 6)*x - (8*n - 24)*x^2)*D[P[n - 1, x], x])/(n - 3)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 0; v[3] = 4; for(n = 4, N, my(i = n); v[i] = (-(3*n - 6 + (8*n^2 - 56*n + 80)*x + (16*n^3 - 160*n^2 + 496*n - 480)*x^2)*v[i - 2] - ((2*n - 4)*x + (8*n^2 - 48*n + 64)*x^2 - (32*n^2 - 160*n + 192)*x^3)*deriv(v[i - 2], x) + (4*n - 11 + (8*n^2 - 56*n + 96)*x)*v[i - 1] + ((2*n - 6)*x - (8*n - 24)*x^2)*deriv(v[i - 1], x))/(n - 3)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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.