Real-rooted atlas

basis-map coefficients of Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1).

Proved for all n: all roots real, in (−∞, 1] · interlacing (weak) what this means

Recurrence
\(\left(-2\,n - n^{2}\right)\,P_{n}(x) = \left(-x - 3\,n x - 2\,n^{2} x\right)\,P_{n-1}(x) + \left(4\,n - 4\,n x + n x^{2} + 4\,n^{2} - 4\,n^{2} x + n^{2} x^{2}\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1022
2-3333
30-121244
410-20-203055
5060-12006066
6-35105105-38510510577
70-280840-280-84039216888

Show all 15 stored rows

Generated b-files: A000027 (row sums) ?

Computed in your browser ?

Technical details
Node id
n-3bc20048ee6f0787 · revision 1 · JSON
Definition source
derived:basis_map_coefficients_of:oeis_catalog:A132812
Fingerprint
f6-4fb3dd87a2a7b214

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): no
  • 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:   0,    2;
  2:  -3,    3,     3;
  3:   0,  -12,    12,    4;
  4:  10,  -20,   -20,   30,     5;
  5:   0,   60,  -120,    0,    60,     6;
  6: -35,  105,   105, -385,   105,   105,    7;
  7:   0, -280,   840, -280,  -840,   392,  168,    8;
  8: 126, -504,  -504, 3276, -2394, -1260, 1008,  252,   9;
  9:   0, 1260, -5040, 3360,  7560, -8820, -840, 2160, 360, 10;
%F Formula
The row polynomials satisfy -(n^2+2*n)*P(n,x) = -(2*n^2+3*n+1)*x*P(n-1,x) + (4*n^2+4*n-(4*n^2+4*n)*x+(n^2+n)*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^2 + 3*n + 1)*x*P[n - 1, x] + (4*n^2 + 4*n - (4*n^2 + 4*n)*x + (n^2 + n)*x^2)*P[n - 2, x])/(-n^2 - 2*n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n^2 + 3*n + 1)*x*v[i - 1] + (4*n^2 + 4*n - (4*n^2 + 4*n)*x + (n^2 + n)*x^2)*v[i - 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 - 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, 1]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-676bb32d6a1b9c64
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-676bb32d6a1b9c64
  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-72ab2c9b35ff70ef

Proof chains

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

Used in 4 proof steps

No candidate equalities or related sequences.

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