Real-rooted atlas

A103327 → basis A013610 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-4\,n + 3\,n x + 8\,n^{2} - 12\,n^{2} x + 12\,n^{3} x\right)\,P_{n}(x) = \left(27\,x^{2} - 16\,n - 108\,n x - 117\,n x^{2} + 32\,n^{2} + 96\,n^{2} x + 48\,n^{3} x + 108\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(6\,x^{2} + 18\,x^{3} - 16\,n x - 72\,n x^{2} - 72\,n x^{3} - 24\,n^{2} x^{2} - 72\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
14610
2167227115
3644806481081300
425626886480432040514149
51024138244838460480243001458149470
640966758431104058060845360012247251031544503
71638431948818247684561920544320029393285715361749615694120

All 15 rows

Technical details
Node id
n-612483c3736b6a28 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-ed1c4c95096a6ecc

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,       6;
  2:     16,      72,       27;
  3:     64,     480,      648,       108;
  4:    256,    2688,     6480,      4320,       405;
  5:   1024,   13824,    48384,     60480,     24300,      1458;
  6:   4096,   67584,   311040,    580608,    453600,    122472,      5103;
  7:  16384,  319488,  1824768,   4561920,   5443200,   2939328,    571536,    17496;
  8:  65536, 1474560, 10063872,  31629312,  51321600,  43110144,  17146080,  2519424,    59049;
  9: 262144, 6684672, 53084160, 201277440, 415134720, 480370176, 301771008, 92378880, 10628820, 196830;
%F Formula
The row polynomials satisfy (8*n^2-4*n+(12*n^3-12*n^2+3*n)*x)*P(n,x) = (32*n^2-16*n+(48*n^3+96*n^2-108*n)*x+(108*n^3-117*n+27)*x^2)*P(n-1,x) - (16*n*x+(24*n^2+72*n-6)*x^2+(72*n^2+72*n-18)*x^3)*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[((32*n^2 - 16*n + (48*n^3 + 96*n^2 - 108*n)*x + (108*n^3 - 117*n + 27)*x^2)*P[n - 1, x] - (16*n*x + (24*n^2 + 72*n - 6)*x^2 + (72*n^2 + 72*n - 18)*x^3)*D[P[n - 1, x], x])/(8*n^2 - 4*n + (12*n^3 - 12*n^2 + 3*n)*x)]];
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] = ((32*n^2 - 16*n + (48*n^3 + 96*n^2 - 108*n)*x + (108*n^3 - 117*n + 27)*x^2)*v[i - 1] - (16*n*x + (24*n^2 + 72*n - 6)*x^2 + (72*n^2 + 72*n - 18)*x^3)*deriv(v[i - 1], x))/(8*n^2 - 4*n + (12*n^3 - 12*n^2 + 3*n)*x)); 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