Real-rooted atlas

A008459 → basis A013609 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-5\,n x + 2\,n x^{2} - 2\,n^{2} + 4\,n^{2} x + 2\,n^{3}\right)\,P_{n}(x) = \left(4 + 14\,x + 8\,x^{2} - 8\,x^{3} - 8\,n - 34\,n x - 26\,n x^{2} + 12\,n x^{3} - 4\,n^{2} + 4\,n^{2} x + 24\,n^{2} x^{2} + 8\,n^{3} + 12\,n^{3} x\right)\,P_{n-1}(x) + \left(2\,x + 6\,x^{2} - 8\,x^{4} - 2\,n x - 12\,n x^{2} - 16\,n x^{3} - 4\,n^{2} x - 8\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1426
21820442
380156728316
43501000840224162430
515125740728036006403218804
6646830744529204032013440172864145684
7274561570803437283612001881604569644801281127928

All 15 rows

Technical details
Node id
n-9591dfb8f9ec5769 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-e96dec9df4b3089a

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:      4,       2;
  2:     18,      20,        4;
  3:     80,     156,       72,        8;
  4:    350,    1000,      840,      224,       16;
  5:   1512,    5740,     7280,     3600,      640,      32;
  6:   6468,   30744,    52920,    40320,    13440,    1728,      64;
  7:  27456,  157080,   343728,   361200,   188160,   45696,    4480,    128;
  8: 115830,  775632,  2066064,  2794176,  2032800,  779520,  145152,  11264,   256;
  9: 486200, 3732300, 11737440, 19507488, 18450432, 9979200, 2956800, 437760, 27648, 512;
%F Formula
The row polynomials satisfy (2*n^3-2*n^2+(4*n^2-5*n)*x+2*n*x^2)*P(n,x) = (8*n^3-4*n^2-8*n+4+(12*n^3+4*n^2-34*n+14)*x+(24*n^2-26*n+8)*x^2+(12*n-8)*x^3)*P(n-1,x) - ((4*n^2+2*n-2)*x+(8*n^2+12*n-6)*x^2+16*n*x^3+8*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[((8*n^3 - 4*n^2 - 8*n + 4 + (12*n^3 + 4*n^2 - 34*n + 14)*x + (24*n^2 - 26*n + 8)*x^2 + (12*n - 8)*x^3)*P[n - 1, x] - ((4*n^2 + 2*n - 2)*x + (8*n^2 + 12*n - 6)*x^2 + 16*n*x^3 + 8*x^4)*D[P[n - 1, x], x])/(2*n^3 - 2*n^2 + (4*n^2 - 5*n)*x + 2*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] = ((8*n^3 - 4*n^2 - 8*n + 4 + (12*n^3 + 4*n^2 - 34*n + 14)*x + (24*n^2 - 26*n + 8)*x^2 + (12*n - 8)*x^3)*v[i - 1] - ((4*n^2 + 2*n - 2)*x + (8*n^2 + 12*n - 6)*x^2 + 16*n*x^3 + 8*x^4)*deriv(v[i - 1], x))/(2*n^3 - 2*n^2 + (4*n^2 - 5*n)*x + 2*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)

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