Real-rooted atlas

A008459 → x ↦ x+1 → basis A013611

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1347
213321661
36322824064595
43211536249615362566145
516831002022080217608960102465527
6898964032178320243200165120491524096712909
7486394033961358448237216022668801139712258048163847863667

All 15 rows

Technical details
Node id
n-ad285e734b9b3fbb · revision 0 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-e88a2f797c4f541a

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:       3,        4;
  2:      13,       32,       16;
  3:      63,      228,      240,        64;
  4:     321,     1536,     2496,      1536,       256;
  5:    1683,    10020,    22080,     21760,      8960,      1024;
  6:    8989,    64032,   178320,    243200,    165120,     49152,      4096;
  7:   48639,   403396,  1358448,   2372160,   2266880,   1139712,    258048,    16384;
  8:  265729,  2514944,  9936640,  21174272,  26199040,  18808832,   7340032,  1310720,   65536;
  9: 1462563, 15554628, 70541568, 177558528, 270240768, 253661184, 143130624, 44826624, 6488064, 262144;
%F Formula
The row polynomials satisfy n*P(n,x) = -(n-1+(8*n-8)*x+(16*n-16)*x^2)*P(n-2,x) + (6*n-3+(8*n-4)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n - 1 + (8*n - 8)*x + (16*n - 16)*x^2)*P[n - 2, x] + (6*n - 3 + (8*n - 4)*x)*P[n - 1, x])/(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] = 3 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n - 1 + (8*n - 8)*x + (16*n - 16)*x^2)*v[i - 2] + (6*n - 3 + (8*n - 4)*x)*v[i - 1])/(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