Real-rooted atlas

A104684 → basis A007318 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2415
32210133
413690181245
58867722342811921
659446430262049040115525
74063652524269106880900541127905
8281488423220261324849101540015127011067925

All 15 rows

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(\left(1 - n\right)\,P_{n}(x) = \left(-8 - 8\,x - 2\,x^{2} + 4\,n + 4\,n x + n x^{2}\right)\,P_{n-2}(x) + \left(12 + 3\,x - 8\,n - 2\,n x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 4 + x\)
Technical details
Node id
n-7fbdda6d907a2282 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-d5b983027f36fb5d)
Fingerprint
f6-a747911520942d51

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,        1;
   3:       22,       10,        1;
   4:      136,       90,       18,        1;
   5:      886,      772,      234,       28,       1;
   6:     5944,     6430,     2620,      490,      40,      1;
   7:    40636,    52524,    26910,     6880,     900,     54,     1;
   8:   281488,   423220,   261324,    84910,   15400,   1512,    70,    1;
   9:  1968934,  3375880,  2439892,   962584,  223930,  30856,  2380,   88,   1;
  10: 13875544, 26720118, 22127256, 10274292, 2921184, 520506, 56952, 3564, 108, 1;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (10*n-16+(3*n-5)*x)*P(n-1,x) - (12+10*x+2*x^2)*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[((10*n - 16 + (3*n - 5)*x)*P[n - 1, x] - (12 + 10*x + 2*x^2)*D[P[n - 1, x], x])/(n - 1)]];
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] = ((10*n - 16 + (3*n - 5)*x)*v[i - 1] - (12 + 10*x + 2*x^2)*deriv(v[i - 1], x))/(n - 1)); 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