Real-rooted atlas

A108838 → reverse → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(3 + 6\,x + 3\,x^{2} + 4\,n + 8\,n x + 4\,n x^{2} + n^{2} + 2\,n^{2} x + n^{2} x^{2}\right)\,P_{n}(x) = \left(3 + 15\,x + 21\,x^{2} + 9\,x^{3} + 4\,n + 17\,n x + 22\,n x^{2} + 9\,n x^{3} + n^{2} + 4\,n^{2} x + 5\,n^{2} x^{2} + 2\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(3\,x + 9\,x^{2} + 9\,x^{3} + 3\,x^{4} + 2\,n x + 6\,n x^{2} + 6\,n x^{3} + 2\,n x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
022
1257
22121428
32215642121
4232144240132550
52453008259904292591
6260550220042904004143012536
72779245005140142102116016486261921

All 15 rows

Technical details
Node id
n-d7c9d6946ca46996 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-63d66b489de6a076

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: 2;
  1: 2,   5;
  2: 2,  12,   14;
  3: 2,  21,   56,    42;
  4: 2,  32,  144,   240,   132;
  5: 2,  45,  300,   825,   990,    429;
  6: 2,  60,  550,  2200,  4290,   4004,   1430;
  7: 2,  77,  924,  5005, 14014,  21021,  16016,   4862;
  8: 2,  96, 1456, 10192, 38220,  81536,  99008,  63648,  16796;
  9: 2, 117, 2184, 19110, 91728, 259896, 445536, 453492, 251940, 58786;
%F Formula
The row polynomials satisfy (n^2+4*n+3+(2*n^2+8*n+6)*x+(n^2+4*n+3)*x^2)*P(n,x) = (n^2+4*n+3+(4*n^2+17*n+15)*x+(5*n^2+22*n+21)*x^2+(2*n^2+9*n+9)*x^3)*P(n-1,x) + ((2*n+3)*x+(6*n+9)*x^2+(6*n+9)*x^3+(2*n+3)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 2.
%t Mathematica
P[0, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + 4*n + 3 + (4*n^2 + 17*n + 15)*x + (5*n^2 + 22*n + 21)*x^2 + (2*n^2 + 9*n + 9)*x^3)*P[n - 1, x] + ((2*n + 3)*x + (6*n + 9)*x^2 + (6*n + 9)*x^3 + (2*n + 3)*x^4)*D[P[n - 1, x], x])/(n^2 + 4*n + 3 + (2*n^2 + 8*n + 6)*x + (n^2 + 4*n + 3)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 2; for(n = 1, N, my(i = n + 1); v[i] = ((n^2 + 4*n + 3 + (4*n^2 + 17*n + 15)*x + (5*n^2 + 22*n + 21)*x^2 + (2*n^2 + 9*n + 9)*x^3)*v[i - 1] + ((2*n + 3)*x + (6*n + 9)*x^2 + (6*n + 9)*x^3 + (2*n + 3)*x^4)*deriv(v[i - 1], x))/(n^2 + 4*n + 3 + (2*n^2 + 8*n + 6)*x + (n^2 + 4*n + 3)*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

Candidate equalities ?

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