Real-rooted atlas

Triangle, T(n, k) = (1/2)*(n+2)! * [x^k]( p(x, n) ), where p(x,0) = 1, p(x,1) = -x, P(x, n) = (1/(n+1))*( (2*n-x)*P(x, n-1) - n*P(x, n-2) ), read by rows.

A136532 (tail)

Proved for all n: all roots real · interlacing what this means

Recurrence from OEIS A136532
\(\left(1 + \frac{1}{2}\,n\right)\,P_{n}(x) = \left(3 - \frac{3}{2}\,x + 4\,n - \frac{1}{2}\,n x + n^{2}\right)\,P_{n-1}(x) + \left(-3 - \frac{11}{2}\,n - 3\,n^{2} - \frac{1}{2}\,n^{3}\right)\,P_{n-2}(x)\)
\(P_{1}(x) = -8 - 16\,x + 4\,x^{2}\)
\(P_{2}(x) = -60 - 65\,x + 50\,x^{2} - 5\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
1-8-164-20
2-60-6550-5-80
3-384-168462-1086-192
4-23807633836-1624196-7784
5-142082124829560-214724256-320819072
6-73836302571199998-26912778840-9387486-9229536
7-1760003761240854530-32889401360150-22888018430-700102299840
8382496444711623-7017472-3941742222743644-5098676568568-33242968-1120282944

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-cfb1b41332b76420 · revision 1 · JSON
Definition source
oeis_catalog:A136532
Fingerprint
f6-f4ff2f177f5d4ad6

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 1)
  • 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 ?

No OEIS matches.

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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:         -8,        -16,           4;
   2:        -60,        -65,          50,          -5;
   3:       -384,       -168,         462,        -108,          6;
   4:      -2380,        763,        3836,       -1624,        196,          -7;
   5:     -14208,      21248,       29560,      -21472,       4256,        -320,         8;
   6:     -73836,     302571,      199998,     -269127,      78840,       -9387,       486,        -9;
   7:    -176000,    3761240,      854530,    -3288940,    1360150,     -228880,     18430,      -700,      10;
   8:    3824964,   44711623,    -7017472,   -39417422,   22743644,    -5098676,    568568,    -33242,     968,    -11;
   9:  104573760,  520004832,  -304428396,  -457688256,  376007784,  -108589488,  15755712,  -1261536,   56184,  -1296,   12;
  10: 1945371428, 5886398167, -6815383510, -4941747993, 6205006704, -2266280562, 411844524, -42381690, 2567292, -90181, 1690, -13;
%F Formula
The row polynomials satisfy ((1/2)*n+1)*P(n,x) = (n^2+4*n+3-((1/2)*n+3/2)*x)*P(n-1,x) - ((1/2)*n^3+3*n^2+(11/2)*n+3)*P(n-2,x) for n >= 3, with P(1,x) = -8 - 16*x + 4*x^2, P(2,x) = -60 - 65*x + 50*x^2 - 5*x^3.
%t Mathematica
P[1, x_] := -8 - 16*x + 4*x^2;
P[2, x_] := -60 - 65*x + 50*x^2 - 5*x^3;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + 4*n + 3 - ((1/2)*n + 3/2)*x)*P[n - 1, x] - ((1/2)*n^3 + 3*n^2 + (11/2)*n + 3)*P[n - 2, x])/((1/2)*n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = -8 - 16*x + 4*x^2; v[2] = -60 - 65*x + 50*x^2 - 5*x^3; for(n = 3, N, my(i = n); v[i] = ((n^2 + 4*n + 3 - ((1/2)*n + 3/2)*x)*v[i - 1] - ((1/2)*n^3 + 3*n^2 + (11/2)*n + 3)*v[i - 2])/((1/2)*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)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 1, 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-8965e6e864ed875e
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (strict, previous interlaces next) for every m >= 2
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-8965e6e864ed875e

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

No candidate equalities or related sequences.

References

  1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023