Real-rooted atlas

A391045 → reverse → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(-2\,n - 2\,n^{2} + 4\,n^{3}\right)\,P_{n}(x) = \left(2\,n + 4\,n x - 13\,n^{2} - 11\,n^{2} x + 11\,n^{3} + 7\,n^{3} x\right)\,P_{n-1}(x) + \left(-6 + 6\,x^{2} + 4\,n - 10\,n x - 14\,n x^{2} + 14\,n^{2} + 16\,n^{2} x + 2\,n^{2} x^{2}\right)\,P_{n-1}'(x) + \left(-6 - 10\,x - 2\,x^{2} + 2\,x^{3} + 15\,n + 25\,n x + 5\,n x^{2} - 5\,n x^{3}\right)\,P_{n-1}''(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2213
366113
4213112165
58015696201353
63227806602303012023
7134738984203208047042112041
85798195092557816485542586156173713

All 15 rows

Technical details
Node id
n-ed28bb67aebf386d · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-727cfd8711a24e2e

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:      2,      1;
   3:      6,      6,      1;
   4:     21,     31,     12,      1;
   5:     80,    156,     96,     20,      1;
   6:    322,    780,    660,    230,     30,      1;
   7:   1347,   3898,   4203,   2080,    470,     42,     1;
   8:   5798,  19509,  25578,  16485,   5425,    861,    56,    1;
   9:  25512,  97848, 151116, 120288,  52500,  12376,  1456,   72,  1;
  10: 114236, 491868, 874548, 829680, 453348, 143892, 25536, 2316, 90, 1;
%F Formula
The row polynomials satisfy (4*n^3-2*n^2-2*n)*P(n,x) = (11*n^3-13*n^2+2*n+(7*n^3-11*n^2+4*n)*x)*P(n-1,x) + (14*n^2+4*n-6+(16*n^2-10*n)*x+(2*n^2-14*n+6)*x^2)*d/dx P(n-1,x) + (15*n-6+(25*n-10)*x+(5*n-2)*x^2-(5*n-2)*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[((11*n^3 - 13*n^2 + 2*n + (7*n^3 - 11*n^2 + 4*n)*x)*P[n - 1, x] + (14*n^2 + 4*n - 6 + (16*n^2 - 10*n)*x + (2*n^2 - 14*n + 6)*x^2)*D[P[n - 1, x], x] + (15*n - 6 + (25*n - 10)*x + (5*n - 2)*x^2 - (5*n - 2)*x^3)*D[P[n - 1, x], {x, 2}])/(4*n^3 - 2*n^2 - 2*n)]];
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] = ((11*n^3 - 13*n^2 + 2*n + (7*n^3 - 11*n^2 + 4*n)*x)*v[i - 1] + (14*n^2 + 4*n - 6 + (16*n^2 - 10*n)*x + (2*n^2 - 14*n + 6)*x^2)*deriv(v[i - 1], x) + (15*n - 6 + (25*n - 10)*x + (5*n - 2)*x^2 - (5*n - 2)*x^3)*deriv(deriv(v[i - 1], x), x))/(4*n^3 - 2*n^2 - 2*n)); 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

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