Real-rooted atlas

A086645 → basis A013611 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(6 - 12\,x - 7\,n + 26\,n x + 2\,n^{2} - 18\,n^{2} x + 4\,n^{3} x\right)\,P_{n}(x) = \left(24 + 96\,x + 32\,x^{2} - 28\,n - 64\,n x + 96\,n x^{2} + 8\,n^{2} - 24\,n^{2} x - 152\,n^{2} x^{2} + 16\,n^{3} x + 48\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(6\,x + 28\,x^{2} + 16\,x^{3} - 4\,n x - 8\,n x^{2} + 32\,n x^{3} - 8\,n^{2} x^{2} - 32\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 2 + 8\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
22810
386448120
4323848642561536
5128204876808192128019328
65121024053760102400640006144237056
72048491523317769175041075200442368286722846720
8819222937618923526881280120422409633792280985613107233628160

All 15 rows

Technical details
Node id
n-4165ce68a3ab41a3 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-b24a5168263ea116

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:      2,       8;
   3:      8,      64,       48;
   4:     32,     384,      864,       256;
   5:    128,    2048,     7680,      8192,      1280;
   6:    512,   10240,    53760,    102400,     64000,       6144;
   7:   2048,   49152,   331776,    917504,   1075200,     442368,      28672;
   8:   8192,  229376,  1892352,   6881280,  12042240,    9633792,    2809856,    131072;
   9:  32768, 1048576, 10223616,  46137344, 108134400,  132120576,   77070336,  16777216,   589824;
  10: 131072, 4718592, 53084160, 286261248, 843448320, 1401421824, 1271660544, 566231040, 95551488, 2621440;
%F Formula
The row polynomials satisfy (2*n^2-7*n+6+(4*n^3-18*n^2+26*n-12)*x)*P(n,x) = (8*n^2-28*n+24+(16*n^3-24*n^2-64*n+96)*x+(48*n^3-152*n^2+96*n+32)*x^2)*P(n-1,x) - ((4*n-6)*x+(8*n^2+8*n-28)*x^2+(32*n^2-32*n-16)*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 8*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^2 - 28*n + 24 + (16*n^3 - 24*n^2 - 64*n + 96)*x + (48*n^3 - 152*n^2 + 96*n + 32)*x^2)*P[n - 1, x] - ((4*n - 6)*x + (8*n^2 + 8*n - 28)*x^2 + (32*n^2 - 32*n - 16)*x^3)*D[P[n - 1, x], x])/(2*n^2 - 7*n + 6 + (4*n^3 - 18*n^2 + 26*n - 12)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 8*x; for(n = 3, N, my(i = n); v[i] = ((8*n^2 - 28*n + 24 + (16*n^3 - 24*n^2 - 64*n + 96)*x + (48*n^3 - 152*n^2 + 96*n + 32)*x^2)*v[i - 1] - ((4*n - 6)*x + (8*n^2 + 8*n - 28)*x^2 + (32*n^2 - 32*n - 16)*x^3)*deriv(v[i - 1], x))/(2*n^2 - 7*n + 6 + (4*n^3 - 18*n^2 + 26*n - 12)*x)); 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

Same fingerprint ?

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