Real-rooted atlas

A257612 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-1 + 2\,n\right)\,P_{n}(x) = \left(4 - 24\,x - 12\,x^{2} + 8\,n - 32\,n x + 8\,n x^{2} + 32\,n^{2} x + 48\,n^{2} x^{2} - 32\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(8\,x - 8\,x^{3} + 16\,n x - 16\,n x^{2} - 32\,n^{2} x^{2} + 32\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-4 + 4\,x - 16\,n x + 16\,n^{2} x\right)\,P_{n-1}(x) + \left(-4\,x + 4\,x^{2} + 8\,n x - 8\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1246
24481264
3836855232960
41624321104048648018432
5321516816147221529637920192430080
6649241620242566028288337376027724844811796480
7128557824232922881329781761662227204658457619523841024371589120

All 15 rows

Technical details
Node id
n-9c65a2e16bd170ad · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-2e37e4aadfbd49d5

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:   2,        4;
  2:   4,       48,         12;
  3:   8,      368,        552,          32;
  4:  16,     2432,      11040,        4864,           80;
  5:  32,    15168,     161472,      215296,        37920,          192;
  6:  64,    92416,    2024256,     6028288,      3373760,       277248,         448;
  7: 128,   557824,   23292288,   132978176,    166222720,     46584576,     1952384,       1024;
  8: 256,  3354624,  254678016,  2544934912,   5984017920,   3817402368,   594248704,   13418496,     2304;
  9: 512, 20145152, 2697738240, 44457926656, 177698032640, 213237639168, 77801371648, 7193968640, 90653184, 5120;
%F Formula
The row polynomials satisfy (2*n-1)*P(n,x) = (8*n+4+(32*n^2-32*n-24)*x-(32*n^3-48*n^2-8*n+12)*x^2)*P(n-2,x) + ((16*n+8)*x-(32*n^2+16*n)*x^2+(32*n^2-8)*x^3)*d/dx P(n-2,x) - (4-(16*n^2-16*n+4)*x)*P(n-1,x) + ((8*n-4)*x-(8*n-4)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((8*n + 4 + (32*n^2 - 32*n - 24)*x - (32*n^3 - 48*n^2 - 8*n + 12)*x^2)*P[n - 2, x] + ((16*n + 8)*x - (32*n^2 + 16*n)*x^2 + (32*n^2 - 8)*x^3)*D[P[n - 2, x], x] - (4 - (16*n^2 - 16*n + 4)*x)*P[n - 1, x] + ((8*n - 4)*x - (8*n - 4)*x^2)*D[P[n - 1, x], x])/(2*n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((8*n + 4 + (32*n^2 - 32*n - 24)*x - (32*n^3 - 48*n^2 - 8*n + 12)*x^2)*v[i - 2] + ((16*n + 8)*x - (32*n^2 + 16*n)*x^2 + (32*n^2 - 8)*x^3)*deriv(v[i - 2], x) - (4 - (16*n^2 - 16*n + 4)*x)*v[i - 1] + ((8*n - 4)*x - (8*n - 4)*x^2)*deriv(v[i - 1], x))/(2*n - 1)); 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. [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.