Real-rooted atlas

A082985 → x ↦ x+1 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-10 - 6\,x + 20\,n + 2\,n x - 2\,n x^{2} + 20\,n^{2} x + 8\,n^{3} x^{2}\right)\,P_{n}(x) = \left(-40 - 74\,x - 36\,x^{2} - 2\,x^{3} + 80\,n + 98\,n x + 4\,n x^{2} - 18\,n x^{3} + 100\,n^{2} x + 120\,n^{2} x^{2} + 32\,n^{2} x^{3} + 32\,n^{3} x^{2} + 24\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-25 - 75\,x - 79\,x^{2} - 33\,x^{3} - 4\,x^{4} - 50\,n x - 130\,n x^{2} - 112\,n x^{3} - 32\,n x^{4} - 20\,n^{2} x^{2} - 36\,n^{2} x^{3} - 16\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
14610
211301556
32911210528274
476378513264451276
5199121021451628550665798
6521374481908216416010149125936
713641131029430367202482591981715120114682

All 15 rows

Technical details
Node id
n-7bb73281baf03735 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-01ea553173440f8c

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:    4,     6;
  2:   11,    30,     15;
  3:   29,   112,    105,     28;
  4:   76,   378,    513,    264,     45;
  5:  199,  1210,   2145,   1628,    550,     66;
  6:  521,  3744,   8190,   8216,   4160,   1014,     91;
  7: 1364, 11310,  29430,  36720,  24825,   9198,   1715,   120;
  8: 3571, 33558, 101235, 151096, 127925,  63852,  18326,  2720,  153;
  9: 9349, 98192, 336927, 585428, 595840, 374718, 146034, 33744, 4104, 190;
%F Formula
The row polynomials satisfy (20*n-10+(20*n^2+2*n-6)*x+(8*n^3-2*n)*x^2)*P(n,x) = (80*n-40+(100*n^2+98*n-74)*x+(32*n^3+120*n^2+4*n-36)*x^2+(24*n^3+32*n^2-18*n-2)*x^3)*P(n-1,x) - (25+(50*n+75)*x+(20*n^2+130*n+79)*x^2+(36*n^2+112*n+33)*x^3+(16*n^2+32*n+4)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((80*n - 40 + (100*n^2 + 98*n - 74)*x + (32*n^3 + 120*n^2 + 4*n - 36)*x^2 + (24*n^3 + 32*n^2 - 18*n - 2)*x^3)*P[n - 1, x] - (25 + (50*n + 75)*x + (20*n^2 + 130*n + 79)*x^2 + (36*n^2 + 112*n + 33)*x^3 + (16*n^2 + 32*n + 4)*x^4)*D[P[n - 1, x], x])/(20*n - 10 + (20*n^2 + 2*n - 6)*x + (8*n^3 - 2*n)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((80*n - 40 + (100*n^2 + 98*n - 74)*x + (32*n^3 + 120*n^2 + 4*n - 36)*x^2 + (24*n^3 + 32*n^2 - 18*n - 2)*x^3)*v[i - 1] - (25 + (50*n + 75)*x + (20*n^2 + 130*n + 79)*x^2 + (36*n^2 + 112*n + 33)*x^3 + (16*n^2 + 32*n + 4)*x^4)*deriv(v[i - 1], x))/(20*n - 10 + (20*n^2 + 2*n - 6)*x + (8*n^3 - 2*n)*x^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

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