Real-rooted atlas

A132812 → basis A007318 → basis A013611

nonnegative coefficients

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

Recurrence

\(\left(2\,n + n^{2}\right)\,P_{n}(x) = \left(-n - 8\,n x - 16\,n x^{2} - n^{2} - 8\,n^{2} x - 16\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(3 + 4\,x + 9\,n + 12\,n x + 6\,n^{2} + 8\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 6 + 8\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
16814
2338448165
31806727682561876
4985484083205760128021185
55418330007584080640384006144239442
629953217644626640902720672000236544286722714173
716634414040324859904885248090316805046272137625613107230868040

All 15 rows

Technical details
Node id
n-979cc2f2c3d8f013 · revision 0 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-f60b2631b48d7d2e

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,        8;
  2:      33,       84,        48;
  3:     180,      672,       768,       256;
  4:     985,     4840,      8320,      5760,       1280;
  5:    5418,    33000,     75840,     80640,      38400,       6144;
  6:   29953,   217644,    626640,    902720,     672000,     236544,     28672;
  7:  166344,  1404032,   4859904,   8852480,    9031680,    5046272,   1376256,    131072;
  8:  927441,  8916624,  36062208,  79511040,  103412736,   79994880,  35094528,   7667712,   589824;
  9: 5188590, 55971720, 259032960, 670817280, 1062512640, 1055416320, 646840320, 230031360, 41287680, 2621440;
%F Formula
The row polynomials satisfy (n^2+2*n)*P(n,x) = -(n^2+n+(8*n^2+8*n)*x+(16*n^2+16*n)*x^2)*P(n-2,x) + (6*n^2+9*n+3+(8*n^2+12*n+4)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 6 + 8*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 6 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 + n + (8*n^2 + 8*n)*x + (16*n^2 + 16*n)*x^2)*P[n - 2, x] + (6*n^2 + 9*n + 3 + (8*n^2 + 12*n + 4)*x)*P[n - 1, x])/(n^2 + 2*n)]];
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 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^2 + n + (8*n^2 + 8*n)*x + (16*n^2 + 16*n)*x^2)*v[i - 2] + (6*n^2 + 9*n + 3 + (8*n^2 + 12*n + 4)*x)*v[i - 1])/(n^2 + 2*n)); 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