Real-rooted atlas

A069466 → basis A007318 → basis A013610

nonnegative coefficients

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

Recurrence

\(\left(n^{2}\right)\,P_{n}(x) = \left(-12 - 72\,x - 108\,x^{2} + 32\,n + 192\,n x + 288\,n x^{2} - 16\,n^{2} - 96\,n^{2} x - 144\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(6 + 6\,x - 24\,n - 24\,n x + 24\,n^{2} + 24\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 6 + 6\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
16612
27814454276
31260342027005407920
4224708064098280453605670252420
5424116189378031298402313360714420612368536752
683058364437417692681820948024004827438010777536673596299889744
71669290481038341304262248386434345911602461619160928215288157621464750578410817307072

All 15 rows

Technical details
Node id
n-647a94c6d9dbf531 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-3d3f3d7068653107

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:           6,            6;
  2:          78,          144,            54;
  3:        1260,         3420,          2700,           540;
  4:       22470,        80640,         98280,         45360,          5670;
  5:      424116,      1893780,       3129840,       2313360,        714420,         61236;
  6:     8305836,     44374176,      92681820,      94802400,      48274380,      10777536,        673596;
  7:   166929048,   1038341304,    2622483864,    3434591160,    2461619160,     928215288,     157621464,      7505784;
  8:  3419932230,  24275496960,   71935063200,  114966371520,  106686379800,   57444266880,   16812956160,   2251735200,    84440070;
  9: 71109813060, 567199510020, 1929223707840, 3642002844480, 4157295302160, 2926680316560, 1238554437120, 290924187840, 31580586180, 956987460;
%F Formula
The row polynomials satisfy n^2*P(n,x) = -(16*n^2-32*n+12+(96*n^2-192*n+72)*x+(144*n^2-288*n+108)*x^2)*P(n-2,x) + (24*n^2-24*n+6+(24*n^2-24*n+6)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 6 + 6*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 6 + 6*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(16*n^2 - 32*n + 12 + (96*n^2 - 192*n + 72)*x + (144*n^2 - 288*n + 108)*x^2)*P[n - 2, x] + (24*n^2 - 24*n + 6 + (24*n^2 - 24*n + 6)*x)*P[n - 1, x])/(n^2)]];
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] = 6 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = (-(16*n^2 - 32*n + 12 + (96*n^2 - 192*n + 72)*x + (144*n^2 - 288*n + 108)*x^2)*v[i - 2] + (24*n^2 - 24*n + 6 + (24*n^2 - 24*n + 6)*x)*v[i - 1])/(n^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

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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023