Real-rooted atlas

A114687 → basis A013609 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(4 + 16\,x + 2\,n + 24\,n x + 16\,n x^{2} + 8\,n^{2} x + 24\,n^{2} x^{2} + 8\,n^{3} x^{2}\right)\,P_{n}(x) = \left(4 + 16\,x + 14\,n + 112\,n x + 208\,n x^{2} + 64\,n x^{3} + 64\,n^{2} x + 264\,n^{2} x^{2} + 224\,n^{2} x^{3} + 56\,n^{3} x^{2} + 96\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-1 - 14\,x - 64\,x^{2} - 112\,x^{3} - 64\,x^{4} - 8\,n x - 68\,n x^{2} - 184\,n x^{3} - 160\,n x^{4} - 8\,n^{2} x^{2} - 48\,n^{2} x^{3} - 64\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
13811
2115648115
3453365762561213
4197193649924608128013013
5903110003792053760320006144141727
6427962184268560515840480000202752286721562287
7207933510081822464442624056448003784704120422413107217385305

All 15 rows

Technical details
Node id
n-cd60910732cfe94d · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-b7bf3ce8c69857a4

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,        8;
  2:     11,       56,       48;
  3:     45,      336,      576,       256;
  4:    197,     1936,     4992,      4608,      1280;
  5:    903,    11000,    37920,     53760,     32000,      6144;
  6:   4279,    62184,   268560,    515840,    480000,    202752,     28672;
  7:  20793,   351008,  1822464,   4426240,   5644800,   3784704,   1204224,    131072;
  8: 103049,  1981472, 12020736,  35338240,  57451520,  53329920,  27295744,   6815744,   589824;
  9: 518859, 11194344, 77709888, 268326912, 531256320, 633249792, 452788224, 184025088, 37158912, 2621440;
%F Formula
The row polynomials satisfy (2*n+4+(8*n^2+24*n+16)*x+(8*n^3+24*n^2+16*n)*x^2)*P(n,x) = (14*n+4+(64*n^2+112*n+16)*x+(56*n^3+264*n^2+208*n)*x^2+(96*n^3+224*n^2+64*n)*x^3)*P(n-1,x) - (1+(8*n+14)*x+(8*n^2+68*n+64)*x^2+(48*n^2+184*n+112)*x^3+(64*n^2+160*n+64)*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[((14*n + 4 + (64*n^2 + 112*n + 16)*x + (56*n^3 + 264*n^2 + 208*n)*x^2 + (96*n^3 + 224*n^2 + 64*n)*x^3)*P[n - 1, x] - (1 + (8*n + 14)*x + (8*n^2 + 68*n + 64)*x^2 + (48*n^2 + 184*n + 112)*x^3 + (64*n^2 + 160*n + 64)*x^4)*D[P[n - 1, x], x])/(2*n + 4 + (8*n^2 + 24*n + 16)*x + (8*n^3 + 24*n^2 + 16*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] = ((14*n + 4 + (64*n^2 + 112*n + 16)*x + (56*n^3 + 264*n^2 + 208*n)*x^2 + (96*n^3 + 224*n^2 + 64*n)*x^3)*v[i - 1] - (1 + (8*n + 14)*x + (8*n^2 + 68*n + 64)*x^2 + (48*n^2 + 184*n + 112)*x^3 + (64*n^2 + 160*n + 64)*x^4)*deriv(v[i - 1], x))/(2*n + 4 + (8*n^2 + 24*n + 16)*x + (8*n^3 + 24*n^2 + 16*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