Real-rooted atlas

A091042 → basis A013610 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(12 - 9\,x - 20\,n + 33\,n x + 8\,n^{2} - 36\,n^{2} x + 12\,n^{3} x\right)\,P_{n}(x) = \left(48 + 180\,x + 108\,x^{2} - 80\,n - 228\,n x - 45\,n x^{2} + 32\,n^{2} - 180\,n^{2} x^{2} + 48\,n^{3} x + 108\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(16\,x + 66\,x^{2} + 54\,x^{3} - 16\,n x - 48\,n x^{2} - 24\,n^{2} x^{2} - 72\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
241822
316120135271
46467215127563004
525634561166412960364531981
6102416896760321330568910016038332146
740967987244928010782721179360530712663393387931
81638436864024883207603200116640008817984285768026244034078648

All 15 rows

Technical details
Node id
n-5a33528b7b7d00cb · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-5246167935ce1568

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:      1;
   2:      4,      18;
   3:     16,     120,      135;
   4:     64,     672,     1512,       756;
   5:    256,    3456,    11664,     12960,      3645;
   6:   1024,   16896,    76032,    133056,     89100,      16038;
   7:   4096,   79872,   449280,   1078272,   1179360,     530712,     66339;
   8:  16384,  368640,  2488320,   7603200,  11664000,    8817984,   2857680,    262440;
   9:  65536, 1671168, 13160448,  48881664,  96940800,  104696064,  58296672,  14276736,  1003833;
  10: 262144, 7471104, 67239936, 294174720, 717050880, 1014114816, 819092736, 351039744, 67315860, 3739770;
%F Formula
The row polynomials satisfy (8*n^2-20*n+12+(12*n^3-36*n^2+33*n-9)*x)*P(n,x) = (32*n^2-80*n+48+(48*n^3-228*n+180)*x+(108*n^3-180*n^2-45*n+108)*x^2)*P(n-1,x) - ((16*n-16)*x+(24*n^2+48*n-66)*x^2+(72*n^2-54)*x^3)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((32*n^2 - 80*n + 48 + (48*n^3 - 228*n + 180)*x + (108*n^3 - 180*n^2 - 45*n + 108)*x^2)*P[n - 1, x] - ((16*n - 16)*x + (24*n^2 + 48*n - 66)*x^2 + (72*n^2 - 54)*x^3)*D[P[n - 1, x], x])/(8*n^2 - 20*n + 12 + (12*n^3 - 36*n^2 + 33*n - 9)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((32*n^2 - 80*n + 48 + (48*n^3 - 228*n + 180)*x + (108*n^3 - 180*n^2 - 45*n + 108)*x^2)*v[i - 1] - ((16*n - 16)*x + (24*n^2 + 48*n - 66)*x^2 + (72*n^2 - 54)*x^3)*deriv(v[i - 1], x))/(8*n^2 - 20*n + 12 + (12*n^3 - 36*n^2 + 33*n - 9)*x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

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