Real-rooted atlas

A391045 → θ+1 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel

Recurrence

\(\left(-2 + 8\,n - 10\,n^{2} + 4\,n^{3}\right)\,P_{n}(x) = \left(-54 - 52\,x + 160\,n + 152\,n x - 142\,n^{2} - 132\,n^{2} x + 40\,n^{3} + 36\,n^{3} x\right)\,P_{n-1}(x) + \left(-54 - 116\,x - 62\,x^{2} + 116\,n + 250\,n x + 134\,n x^{2} - 44\,n^{2} - 96\,n^{2} x - 52\,n^{2} x^{2}\right)\,P_{n-1}'(x) + \left(-9 - 30\,x - 33\,x^{2} - 12\,x^{3} + 15\,n + 50\,n x + 55\,n x^{2} + 20\,n x^{3}\right)\,P_{n-1}''(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2325
31214329
45390424189
524454439610051289
6115231803150132020569013
755311820422830134603675378764085
8268751027181559461188604711089886448461149

All 15 rows

Technical details
Node id
n-cabbdb9e098455ef · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-a33a1afd14740aa7

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:      3,       2;
   3:     12,      14,       3;
   4:     53,      90,      42,       4;
   5:    244,     544,     396,     100,       5;
   6:   1152,    3180,    3150,    1320,     205,       6;
   7:   5531,   18204,   22830,   13460,    3675,     378,      7;
   8:  26875,  102718,  155946,  118860,   47110,    8988,    644,     8;
   9: 131760,  573528, 1022364,  956928,  500080,  142632,  19908,  1032,    9;
  10: 650492, 3176712, 6503328, 7222320, 4698540, 1800288, 386148, 40752, 1575, 10;
%F Formula
The row polynomials satisfy (4*n^3-10*n^2+8*n-2)*P(n,x) = (40*n^3-142*n^2+160*n-54+(36*n^3-132*n^2+152*n-52)*x)*P(n-1,x) - (44*n^2-116*n+54+(96*n^2-250*n+116)*x+(52*n^2-134*n+62)*x^2)*d/dx P(n-1,x) + (15*n-9+(50*n-30)*x+(55*n-33)*x^2+(20*n-12)*x^3)*d^2/dx^2 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[((40*n^3 - 142*n^2 + 160*n - 54 + (36*n^3 - 132*n^2 + 152*n - 52)*x)*P[n - 1, x] - (44*n^2 - 116*n + 54 + (96*n^2 - 250*n + 116)*x + (52*n^2 - 134*n + 62)*x^2)*D[P[n - 1, x], x] + (15*n - 9 + (50*n - 30)*x + (55*n - 33)*x^2 + (20*n - 12)*x^3)*D[P[n - 1, x], {x, 2}])/(4*n^3 - 10*n^2 + 8*n - 2)]];
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] = ((40*n^3 - 142*n^2 + 160*n - 54 + (36*n^3 - 132*n^2 + 152*n - 52)*x)*v[i - 1] - (44*n^2 - 116*n + 54 + (96*n^2 - 250*n + 116)*x + (52*n^2 - 134*n + 62)*x^2)*deriv(v[i - 1], x) + (15*n - 9 + (50*n - 30)*x + (55*n - 33)*x^2 + (20*n - 12)*x^3)*deriv(deriv(v[i - 1], x), x))/(4*n^3 - 10*n^2 + 8*n - 2)); 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