Real-rooted atlas

A008292 → γ (scale 1/4) → x ↦ x+1 → D

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel

Recurrence

\(\left(-3 + n\right)\,P_{n}(x) = \left(6 + 20\,x + 30\,x^{2} - 11\,n - 34\,n x - 31\,n x^{2} + 6\,n^{2} + 16\,n^{2} x + 10\,n^{2} x^{2} - n^{3} - 2\,n^{3} x - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(4\,x + 16\,x^{2} + 12\,x^{3} - 6\,n x - 16\,n x^{2} - 10\,n x^{3} + 2\,n^{2} x + 4\,n^{2} x^{2} + 2\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(8 + 24\,x - 8\,n - 14\,n x + 2\,n^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(6\,x + 6\,x^{2} - 2\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 0\)
\(P_{3}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
10
20
322
41616
512032152
69605441504
78400739281616608
8806409676823808201216
98467201290240506880317442675584
109676800179020809768960141516838763008
11119750400260789760183110400433213441768960608740864
12159667200040023244803445770240114987417611184128010306482176
13228324096006476101632066144798720287211724804647552000134209536187241158656
1434871316480011042583552001306290585600701485424640162196070400114225438723634366144512
15566658892800019818531532800266641030656001710244343808051952642560006181734850561332630118475078431006720
Technical details
Node id
n-b1e51615932ea943 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-1ffc6ed6313caee0

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:       2;
   4:      16;
   5:     120,       32;
   6:     960,      544;
   7:    8400,     7392,     816;
   8:   80640,    96768,   23808;
   9:  846720,  1290240,  506880,   31744;
  10: 9676800, 17902080, 9768960, 1415168;
%F Formula
The row polynomials satisfy (n-3)*P(n,x) = -(n^3-6*n^2+11*n-6+(2*n^3-16*n^2+34*n-20)*x+(n^3-10*n^2+31*n-30)*x^2)*P(n-2,x) + ((2*n^2-6*n+4)*x+(4*n^2-16*n+16)*x^2+(2*n^2-10*n+12)*x^3)*d/dx P(n-2,x) + (2*n^2-8*n+8+(2*n^2-14*n+24)*x)*P(n-1,x) - ((2*n-6)*x+(2*n-6)*x^2)*d/dx P(n-1,x) for n >= 4, with P(1,x) = 0, P(2,x) = 0, P(3,x) = 2.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 0;
P[3, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - 6*n^2 + 11*n - 6 + (2*n^3 - 16*n^2 + 34*n - 20)*x + (n^3 - 10*n^2 + 31*n - 30)*x^2)*P[n - 2, x] + ((2*n^2 - 6*n + 4)*x + (4*n^2 - 16*n + 16)*x^2 + (2*n^2 - 10*n + 12)*x^3)*D[P[n - 2, x], x] + (2*n^2 - 8*n + 8 + (2*n^2 - 14*n + 24)*x)*P[n - 1, x] - ((2*n - 6)*x + (2*n - 6)*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] = 2; for(n = 4, N, my(i = n); v[i] = (-(n^3 - 6*n^2 + 11*n - 6 + (2*n^3 - 16*n^2 + 34*n - 20)*x + (n^3 - 10*n^2 + 31*n - 30)*x^2)*v[i - 2] + ((2*n^2 - 6*n + 4)*x + (4*n^2 - 16*n + 16)*x^2 + (2*n^2 - 10*n + 12)*x^3)*deriv(v[i - 2], x) + (2*n^2 - 8*n + 8 + (2*n^2 - 14*n + 24)*x)*v[i - 1] - ((2*n - 6)*x + (2*n - 6)*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.