Real-rooted atlas

A111006 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(2\,n + 3\,n^{2} + 2\,n^{2} x - 5\,n^{2} x^{2} + n^{3} + 4\,n^{3} x + 5\,n^{3} x^{2}\right)\,P_{n}(x) = \left(4 + 28\,x + 44\,x^{2} + 20\,x^{3} + 2\,n + 20\,n x + 53\,n x^{2} + 60\,n x^{3} + 25\,n x^{4} - 4\,n^{2} - 22\,n^{2} x - 46\,n^{2} x^{2} - 30\,n^{2} x^{3} - 2\,n^{3} - 14\,n^{3} x - 39\,n^{3} x^{2} - 50\,n^{3} x^{3} - 25\,n^{3} x^{4}\right)\,P_{n-1}(x) + \left(8\,x + 26\,x^{2} + 28\,x^{3} + 10\,x^{4} + 12\,n x + 59\,n x^{2} + 107\,n x^{3} + 85\,n x^{4} + 25\,n x^{5} + 4\,n^{2} x + 25\,n^{2} x^{2} + 61\,n^{2} x^{3} + 65\,n^{2} x^{4} + 25\,n^{2} x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1022
20268
30061218
4003202548
5000125048110
600044512091260
7000020132266168586
800005843575683061320
9000003028088811705502918
10000006140840209723509796412
1100000042520230447404620172813954
12000000721617105940103628928302930192
130000000568825110145642203217004527864926
14000000083153150142453432045786319909150138964
Technical details
Node id
n-3ac4a93aea3ff306 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-9642481e989ba692

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • 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: 1;
  1: 0, 2;
  2: 0, 2, 6;
  3: 0, 0, 6, 12;
  4: 0, 0, 3, 20, 25;
  5: 0, 0, 0, 12, 50,  48;
  6: 0, 0, 0,  4, 45, 120,  91;
  7: 0, 0, 0,  0, 20, 132, 266, 168;
  8: 0, 0, 0,  0,  5,  84, 357, 568,  306;
  9: 0, 0, 0,  0,  0,  30, 280, 888, 1170, 550;
%F Formula
The row polynomials satisfy (n^3+3*n^2+2*n+(4*n^3+2*n^2)*x+(5*n^3-5*n^2)*x^2)*P(n,x) = -(2*n^3+4*n^2-2*n-4+(14*n^3+22*n^2-20*n-28)*x+(39*n^3+46*n^2-53*n-44)*x^2+(50*n^3+30*n^2-60*n-20)*x^3+(25*n^3-25*n)*x^4)*P(n-1,x) + ((4*n^2+12*n+8)*x+(25*n^2+59*n+26)*x^2+(61*n^2+107*n+28)*x^3+(65*n^2+85*n+10)*x^4+(25*n^2+25*n)*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^3 + 4*n^2 - 2*n - 4 + (14*n^3 + 22*n^2 - 20*n - 28)*x + (39*n^3 + 46*n^2 - 53*n - 44)*x^2 + (50*n^3 + 30*n^2 - 60*n - 20)*x^3 + (25*n^3 - 25*n)*x^4)*P[n - 1, x] + ((4*n^2 + 12*n + 8)*x + (25*n^2 + 59*n + 26)*x^2 + (61*n^2 + 107*n + 28)*x^3 + (65*n^2 + 85*n + 10)*x^4 + (25*n^2 + 25*n)*x^5)*D[P[n - 1, x], x])/(n^3 + 3*n^2 + 2*n + (4*n^3 + 2*n^2)*x + (5*n^3 - 5*n^2)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(2*n^3 + 4*n^2 - 2*n - 4 + (14*n^3 + 22*n^2 - 20*n - 28)*x + (39*n^3 + 46*n^2 - 53*n - 44)*x^2 + (50*n^3 + 30*n^2 - 60*n - 20)*x^3 + (25*n^3 - 25*n)*x^4)*v[i - 1] + ((4*n^2 + 12*n + 8)*x + (25*n^2 + 59*n + 26)*x^2 + (61*n^2 + 107*n + 28)*x^3 + (65*n^2 + 85*n + 10)*x^4 + (25*n^2 + 25*n)*x^5)*deriv(v[i - 1], x))/(n^3 + 3*n^2 + 2*n + (4*n^3 + 2*n^2)*x + (5*n^3 - 5*n^2)*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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.