Real-rooted atlas

A033282 → basis A013610 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-24 - 12\,n + 54\,n x^{2} + 81\,n^{2} x^{2} + 27\,n^{3} x^{2}\right)\,P_{n}(x) = \left(-72 - 216\,x + 324\,x^{3} - 36\,n - 180\,n x + 162\,n x^{2} + 810\,n x^{3} - 36\,n^{2} x + 243\,n^{2} x^{2} + 648\,n^{2} x^{3} + 81\,n^{3} x^{2} + 162\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-8 - 36\,x + 162\,x^{3} + 162\,x^{4} + 72\,n x^{2} + 324\,n x^{3} + 324\,n x^{4} + 36\,n^{2} x^{2} + 162\,n^{2} x^{3} + 162\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
131215
21190135236
345558170115123816
4197327615120272161701062819
5903188401166403175204009501924561047309
64279107406834705304722058137755629338218918717625910
72079361005057068552615382066891825977836867662154525019280298807854

All 15 rows

Technical details
Node id
n-7140bc220af5353e · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-960e5d4ca56c7c6f

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,       12;
  2:     11,       90,       135;
  3:     45,      558,      1701,       1512;
  4:    197,     3276,     15120,      27216,      17010;
  5:    903,    18840,    116640,     317520,     400950,      192456;
  6:   4279,   107406,    834705,    3047220,    5813775,     5629338,     2189187;
  7:  20793,   610050,   5706855,   26153820,   66891825,    97783686,    76621545,    25019280;
  8: 103049,  3460248,  37854432,  208864656,  670539870,  1313512200,  1552863312,  1020786624,   287096238;
  9: 518859, 19622628, 245795472, 1586357136, 6131713770, 15074343648, 23872487184, 23663689920, 13389124554, 3305956680;
%F Formula
The row polynomials satisfy -(12*n+24-(27*n^3+81*n^2+54*n)*x^2)*P(n,x) = -(36*n+72+(36*n^2+180*n+216)*x-(81*n^3+243*n^2+162*n)*x^2-(162*n^3+648*n^2+810*n+324)*x^3)*P(n-1,x) - (8+36*x-(36*n^2+72*n)*x^2-(162*n^2+324*n+162)*x^3-(162*n^2+324*n+162)*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[(-(36*n + 72 + (36*n^2 + 180*n + 216)*x - (81*n^3 + 243*n^2 + 162*n)*x^2 - (162*n^3 + 648*n^2 + 810*n + 324)*x^3)*P[n - 1, x] - (8 + 36*x - (36*n^2 + 72*n)*x^2 - (162*n^2 + 324*n + 162)*x^3 - (162*n^2 + 324*n + 162)*x^4)*D[P[n - 1, x], x])/(-12*n - 24 + (27*n^3 + 81*n^2 + 54*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] = (-(36*n + 72 + (36*n^2 + 180*n + 216)*x - (81*n^3 + 243*n^2 + 162*n)*x^2 - (162*n^3 + 648*n^2 + 810*n + 324)*x^3)*v[i - 1] - (8 + 36*x - (36*n^2 + 72*n)*x^2 - (162*n^2 + 324*n + 162)*x^3 - (162*n^2 + 324*n + 162)*x^4)*deriv(v[i - 1], x))/(-12*n - 24 + (27*n^3 + 81*n^2 + 54*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