Real-rooted atlas

A146986 → basis A007318

nonnegative coefficients

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

Recurrence

\(\left(-3 + n\right)\,P_{n}(x) = \left(24 + 24\,x + 6\,x^{2} - 8\,n - 8\,n x - 2\,n x^{2}\right)\,P_{n-2}(x) + \left(-18 - 9\,x + 6\,n + 3\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + x\)
\(P_{2}(x) = 6 + 6\,x + x^{2}\)
\(P_{3}(x) = 16 + 24\,x + 10\,x^{2} + x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1213
266113
3162410151
4489664161225
51604003681522611107
65761728203211843484415913
721767616109128240348078878133291
8844833792570885299229280966417761441193185
9332801497602913283207682190729520025760398427411139427
1013209666048014533121850368150438481088029008066496888453216777513
115263362894848710604810265856967603262227202767296840864167208196761046140487931
122101248126074883414630455162880591091204414156823455488887270423426404113604327220721242396145
138396800545792001616404482886512643463961602944550401818854408229196827074784631928099352094520412211452782547
1433570816234995712755347456147714048019618621441869587456131568307269249945627269088079362624165966242362208205164822018711912313
Technical details
Node id
n-a87dc3a277e666f3 · revision 0 · JSON
Definition source
derived:basis_map_A007318
Fingerprint
f6-e3f85018fd82ff08

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,      1;
  2:     6,      6,      1;
  3:    16,     24,     10,      1;
  4:    48,     96,     64,     16,      1;
  5:   160,    400,    368,    152,     26,     1;
  6:   576,   1728,   2032,   1184,    348,    44,     1;
  7:  2176,   7616,  10912,   8240,   3480,   788,    78,    1;
  8:  8448,  33792,  57088,  52992,  29280,  9664,  1776,  144,   1;
  9: 33280, 149760, 291328, 320768, 219072, 95200, 25760, 3984, 274, 1;
%F Formula
The row polynomials satisfy (n-3)*P(n,x) = -(8*n-24+(8*n-24)*x+(2*n-6)*x^2)*P(n-2,x) + (6*n-18+(3*n-9)*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + x, P(2,x) = 6 + 6*x + x^2, P(3,x) = 16 + 24*x + 10*x^2 + x^3.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + x;
P[2, x_] := 6 + 6*x + x^2;
P[3, x_] := 16 + 24*x + 10*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(8*n - 24 + (8*n - 24)*x + (2*n - 6)*x^2)*P[n - 2, x] + (6*n - 18 + (3*n - 9)*x)*P[n - 1, x])/(n - 3)]];
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 + x; v[3] = 6 + 6*x + x^2; v[4] = 16 + 24*x + 10*x^2 + x^3; for(n = 4, N, my(i = n + 1); v[i] = (-(8*n - 24 + (8*n - 24)*x + (2*n - 6)*x^2)*v[i - 2] + (6*n - 18 + (3*n - 9)*x)*v[i - 1])/(n - 3)); 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. [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