Real-rooted atlas

A011973 → ⊙ C(n,k) → D+1

nonnegative coefficients

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence
\(\left(3\,n - 11\,n x + 4\,n x^{2} - 3\,n^{2} + 4\,n^{2} x + n^{3}\right)\,P_{n}(x) = \left(28\,x^{2} - 16\,x^{3} + n + n x - 40\,n x^{2} + 16\,n x^{3} - n^{2} - 12\,n^{2} x + 16\,n^{2} x^{2} + n^{3} + 4\,n^{3} x\right)\,P_{n-1}(x) + \left(-6\,x^{2} + 32\,x^{3} - 32\,x^{4} - 2\,n x + 16\,n x^{2} - 32\,n x^{3} + 2\,n^{2} x - 8\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
011
111
2325
37613
41324643
5218030131
63121015020411
7434626301401275

Show all 15 stored rows

Generated b-files: A391990 (row sums) ?

Computed in your browser ?

Technical details
Node id
n-e21aec9025eacedd · revision 1 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-4806a24fb0d130c1

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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

OEIS matches ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:  1;
  1:  1;
  2:  3,    2;
  3:  7,    6;
  4: 13,   24,    6;
  5: 21,   80,   30;
  6: 31,  210,  150,   20;
  7: 43,  462,  630,  140;
  8: 57,  896, 2100,  840,  70;
  9: 73, 1584, 5796, 4200, 630;
%F Formula
The row polynomials satisfy (n^3-3*n^2+3*n+(4*n^2-11*n)*x+4*n*x^2)*P(n,x) = (n^3-n^2+n+(4*n^3-12*n^2+n)*x+(16*n^2-40*n+28)*x^2+(16*n-16)*x^3)*P(n-1,x) + ((2*n^2-2*n)*x-(8*n^2-16*n+6)*x^2-(32*n-32)*x^3-32*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[((n^3 - n^2 + n + (4*n^3 - 12*n^2 + n)*x + (16*n^2 - 40*n + 28)*x^2 + (16*n - 16)*x^3)*P[n - 1, x] + ((2*n^2 - 2*n)*x - (8*n^2 - 16*n + 6)*x^2 - (32*n - 32)*x^3 - 32*x^4)*D[P[n - 1, x], x])/(n^3 - 3*n^2 + 3*n + (4*n^2 - 11*n)*x + 4*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] = ((n^3 - n^2 + n + (4*n^3 - 12*n^2 + n)*x + (16*n^2 - 40*n + 28)*x^2 + (16*n - 16)*x^3)*v[i - 1] + ((2*n^2 - 2*n)*x - (8*n^2 - 16*n + 6)*x^2 - (32*n - 32)*x^3 - 32*x^4)*deriv(v[i - 1], x))/(n^3 - 3*n^2 + 3*n + (4*n^2 - 11*n)*x + 4*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)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-dd3a7d27a8ae2c9a
  • Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
    All coefficients of each polynomial have the same sign (after x -> -x when alternating).
    derived from the cited steps by a stated theorem: e-dd3a7d27a8ae2c9a
  • Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
    Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
    derived from the cited steps by a stated theorem: e-dd3a7d27a8ae2c9a

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

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