Real-rooted atlas

A091044 → basis A007318 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(2\,n + 4\,n^{2} + n^{2} x + 2\,n^{3} x\right)\,P_{n}(x) = \left(-4\,x - x^{2} + 8\,n + 4\,n x - 2\,n x^{2} + 16\,n^{2} + 28\,n^{2} x + 9\,n^{2} x^{2} + 8\,n^{3} x + 6\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-4\,x - 6\,x^{2} - 2\,x^{3} - 8\,n x - 16\,n x^{2} - 8\,n x^{3} - 4\,n^{2} x^{2} - 4\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1448
21632957
36419212016392
425610241008320252633
510245120691235847003617376
6409624576422403072010080134449113105
71638411468823961622528010560024192235264728176
86553652428812902401490944915200304128517443840814646001
926214423592966684672917504069888003075072768768101376594010029421208
1010485761048576033619968534773764874240026836992896896017571841853288800121185131465
1141943044613734416515072029884416031752192021056716888711168234291203706560320320125841441158595512
1216777216201326592795869184161480704019611648001524105216777928704260702208560102407321600528528174721697216558969
136710886487241523237748736008489271296116064256001035495014462234296322540175360696563712124467200136913928386562366019644764234544
1426843545637580963841766640844843620761600663224320006685301145646309638144223535431687501455360171991040026013644824440832128710431360225276659588385
Technical details
Node id
n-a4ad45001919942d · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-d5443f71e0d20378

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,       4;
  2:     16,      32,       9;
  3:     64,     192,     120,      16;
  4:    256,    1024,    1008,     320,      25;
  5:   1024,    5120,    6912,    3584,     700,      36;
  6:   4096,   24576,   42240,   30720,   10080,    1344,     49;
  7:  16384,  114688,  239616,  225280,  105600,   24192,   2352,     64;
  8:  65536,  524288, 1290240, 1490944,  915200,  304128,  51744,   3840,   81;
  9: 262144, 2359296, 6684672, 9175040, 6988800, 3075072, 768768, 101376, 5940, 100;
%F Formula
The row polynomials satisfy (4*n^2+2*n+(2*n^3+n^2)*x)*P(n,x) = (16*n^2+8*n+(8*n^3+28*n^2+4*n-4)*x+(6*n^3+9*n^2-2*n-1)*x^2)*P(n-1,x) - ((8*n+4)*x+(4*n^2+16*n+6)*x^2+(4*n^2+8*n+2)*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[((16*n^2 + 8*n + (8*n^3 + 28*n^2 + 4*n - 4)*x + (6*n^3 + 9*n^2 - 2*n - 1)*x^2)*P[n - 1, x] - ((8*n + 4)*x + (4*n^2 + 16*n + 6)*x^2 + (4*n^2 + 8*n + 2)*x^3)*D[P[n - 1, x], x])/(4*n^2 + 2*n + (2*n^3 + n^2)*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] = ((16*n^2 + 8*n + (8*n^3 + 28*n^2 + 4*n - 4)*x + (6*n^3 + 9*n^2 - 2*n - 1)*x^2)*v[i - 1] - ((8*n + 4)*x + (4*n^2 + 16*n + 6)*x^2 + (4*n^2 + 8*n + 2)*x^3)*deriv(v[i - 1], x))/(4*n^2 + 2*n + (2*n^3 + n^2)*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