Real-rooted atlas

A367270 → x ↦ x+1 → basis A013609

nonnegative coefficients

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

Recurrence

\(\left(-3\,n + 2\,n^{2}\right)\,P_{n}(x) = \left(-2 - 8\,x - 8\,x^{2} + 5\,n + 20\,n x + 20\,n x^{2} - 2\,n^{2} - 8\,n^{2} x - 8\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(10 + 8\,x - 24\,n - 16\,n x + 12\,n^{2} + 8\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
011
111
2549
325361273
412926416832593
568118001680640804881
6365311820145208320216019240665
7198257585211550088480352806720448342105

All 15 rows

Technical details
Node id
n-9ac4272ad66435ce · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-06e426b054a33769

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
  • 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:      1;
  2:      5,       4;
  3:     25,      36,      12;
  4:    129,     264,     168,      32;
  5:    681,    1800,    1680,     640,      80;
  6:   3653,   11820,   14520,    8320,    2160,     192;
  7:  19825,   75852,  115500,   88480,   35280,    6720,    448;
  8: 108545,  479248,  870576,  835520,  451360,  134400,  19712,  1024;
  9: 598417, 2994192, 6318144, 7289856, 4979520, 2032128, 473088, 55296, 2304;
%F Formula
The row polynomials satisfy (2*n^2-3*n)*P(n,x) = -(2*n^2-5*n+2+(8*n^2-20*n+8)*x+(8*n^2-20*n+8)*x^2)*P(n-2,x) + (12*n^2-24*n+10+(8*n^2-16*n+8)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^2 - 5*n + 2 + (8*n^2 - 20*n + 8)*x + (8*n^2 - 20*n + 8)*x^2)*P[n - 2, x] + (12*n^2 - 24*n + 10 + (8*n^2 - 16*n + 8)*x)*P[n - 1, x])/(2*n^2 - 3*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] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n^2 - 5*n + 2 + (8*n^2 - 20*n + 8)*x + (8*n^2 - 20*n + 8)*x^2)*v[i - 2] + (12*n^2 - 24*n + 10 + (8*n^2 - 16*n + 8)*x)*v[i - 1])/(2*n^2 - 3*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