Real-rooted atlas

A063007 → basis A013610 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-75 - 27\,x + 27\,x^{2} + 156\,n + 27\,n x - 27\,n x^{2} - 108\,n^{2} + 27\,n^{3}\right)\,P_{n}(x) = \left(-27 - 45\,x + 81\,x^{2} + 162\,x^{3} + 108\,n + 207\,n x - 81\,n x^{2} - 162\,n x^{3} - 162\,n^{2} - 324\,n^{2} x + 81\,n^{3} + 162\,n^{3} x\right)\,P_{n-1}(x) + \left(28 + 90\,x - 72\,x^{2} - 324\,x^{3} - 162\,x^{4} - 72\,n - 324\,n x - 324\,n x^{2} + 36\,n^{2} + 162\,n^{2} x + 162\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
29615
36716254283
4459201624305405445
530211944042390340205670104541
619413165150527310763560459270612361995939
7122767130069854923401163484012604410606236467359637891015
8767511974097651254532143564400227338650196690032788107327505784715672617

All 15 rows

Technical details
Node id
n-af48e301204acbca · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-3160782481402e19

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:        1;
   2:        9,         6;
   3:       67,       162,         54;
   4:      459,      2016,       2430,         540;
   5:     3021,     19440,      42390,       34020,        5670;
   6:    19413,    165150,     527310,      763560,      459270,       61236;
   7:   122767,   1300698,    5492340,    11634840,    12604410,     6062364,      673596;
   8:   767511,   9740976,   51254532,   143564400,   227338650,   196690032,    78810732,     7505784;
   9:  4757017,  70376688,  443836260,  1549270800,  3242803410,  4098158064,  2951024076,  1013280840,    84440070;
  10: 29285793, 494956710, 3640465620, 15233997240, 39733744050, 66349879596, 69747497820, 43008142320, 12919330710, 956987460;
%F Formula
The row polynomials satisfy (27*n^3-108*n^2+156*n-75+(27*n-27)*x-(27*n-27)*x^2)*P(n,x) = (81*n^3-162*n^2+108*n-27+(162*n^3-324*n^2+207*n-45)*x-(81*n-81)*x^2-(162*n-162)*x^3)*P(n-1,x) + (36*n^2-72*n+28+(162*n^2-324*n+90)*x+(162*n^2-324*n-72)*x^2-324*x^3-162*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((81*n^3 - 162*n^2 + 108*n - 27 + (162*n^3 - 324*n^2 + 207*n - 45)*x - (81*n - 81)*x^2 - (162*n - 162)*x^3)*P[n - 1, x] + (36*n^2 - 72*n + 28 + (162*n^2 - 324*n + 90)*x + (162*n^2 - 324*n - 72)*x^2 - 324*x^3 - 162*x^4)*D[P[n - 1, x], x])/(27*n^3 - 108*n^2 + 156*n - 75 + (27*n - 27)*x - (27*n - 27)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((81*n^3 - 162*n^2 + 108*n - 27 + (162*n^3 - 324*n^2 + 207*n - 45)*x - (81*n - 81)*x^2 - (162*n - 162)*x^3)*v[i - 1] + (36*n^2 - 72*n + 28 + (162*n^2 - 324*n + 90)*x + (162*n^2 - 324*n - 72)*x^2 - 324*x^3 - 162*x^4)*deriv(v[i - 1], x))/(27*n^3 - 108*n^2 + 156*n - 75 + (27*n - 27)*x - (27*n - 27)*x^2)); 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)

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