Real-rooted atlas

A232535 → basis A013610 → θ+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(-2 - 3\,x + 6\,n + 2\,n x - 4\,n^{2} + 11\,n^{2} x + 3\,n^{2} x^{2} - 6\,n^{3} x + 6\,n^{3} x^{2}\right)\,P_{n}(x) = \left(-8 - 44\,x - 54\,x^{2} - 9\,x^{3} + 24\,n + 108\,n x + 90\,n x^{2} - 18\,n x^{3} - 16\,n^{2} - 28\,n^{2} x + 69\,n^{2} x^{2} + 81\,n^{2} x^{3} - 24\,n^{3} x - 30\,n^{3} x^{2} + 54\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-4\,x - 22\,x^{2} - 36\,x^{3} - 18\,x^{4} + 8\,n x + 32\,n x^{2} - 72\,n x^{4} + 12\,n^{2} x^{2} + 24\,n^{2} x^{3} - 36\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
131215
2128481177
3484809994321959
4192249679928208202520913
5768122885313688128546758748217743
6307258368317952732672759780317844357212225409
7122882703361776384525312077371205552064167888713996822420167

All 15 rows

Technical details
Node id
n-3087d085e780a0cc · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-6ea98109f528f932

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:      1;
  1:      3,      12;
  2:     12,      84,       81;
  3:     48,     480,      999,       432;
  4:    192,    2496,     7992,      8208,      2025;
  5:    768,   12288,    53136,     88128,     54675,      8748;
  6:   3072,   58368,   317952,    732672,    759780,    317844,     35721;
  7:  12288,  270336,  1776384,   5253120,   7737120,   5552064,   1678887,    139968;
  8:  49152, 1228800,  9455616,  34172928,  65577600,  68024448,  36006768,   8258112,   531441;
  9: 196608, 5505024, 48549888, 207470592, 491961600, 673339392, 522710496, 213311232, 38440899, 1968300;
%F Formula
The row polynomials satisfy -(4*n^2-6*n+2+(6*n^3-11*n^2-2*n+3)*x-(6*n^3+3*n^2)*x^2)*P(n,x) = -(16*n^2-24*n+8+(24*n^3+28*n^2-108*n+44)*x+(30*n^3-69*n^2-90*n+54)*x^2-(54*n^3+81*n^2-18*n-9)*x^3)*P(n-1,x) + ((8*n-4)*x+(12*n^2+32*n-22)*x^2+(24*n^2-36)*x^3-(36*n^2+72*n+18)*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[(-(16*n^2 - 24*n + 8 + (24*n^3 + 28*n^2 - 108*n + 44)*x + (30*n^3 - 69*n^2 - 90*n + 54)*x^2 - (54*n^3 + 81*n^2 - 18*n - 9)*x^3)*P[n - 1, x] + ((8*n - 4)*x + (12*n^2 + 32*n - 22)*x^2 + (24*n^2 - 36)*x^3 - (36*n^2 + 72*n + 18)*x^4)*D[P[n - 1, x], x])/(-4*n^2 + 6*n - 2 - (6*n^3 - 11*n^2 - 2*n + 3)*x + (6*n^3 + 3*n^2)*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] = (-(16*n^2 - 24*n + 8 + (24*n^3 + 28*n^2 - 108*n + 44)*x + (30*n^3 - 69*n^2 - 90*n + 54)*x^2 - (54*n^3 + 81*n^2 - 18*n - 9)*x^3)*v[i - 1] + ((8*n - 4)*x + (12*n^2 + 32*n - 22)*x^2 + (24*n^2 - 36)*x^3 - (36*n^2 + 72*n + 18)*x^4)*deriv(v[i - 1], x))/(-4*n^2 + 6*n - 2 - (6*n^3 - 11*n^2 - 2*n + 3)*x + (6*n^3 + 3*n^2)*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

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