Real-rooted atlas

A126216 → D → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-2 - 3\,n + 2\,n x + 3\,n^{2} - 3\,n^{2} x + 2\,n^{3} + n^{4} x\right)\,P_{n}(x) = \left(-4\,x + 2\,x^{2} + 2\,n - 10\,n x - n x^{2} + 8\,n^{2} - 4\,n^{2} x^{2} + 8\,n^{3} + 10\,n^{3} x + 4\,n^{4} x + 3\,n^{4} x^{2}\right)\,P_{n-1}(x) + \left(-x + x^{3} - 4\,n x - 4\,n x^{2} - 4\,n^{2} x - 6\,n^{2} x^{2} - 2\,n^{2} x^{3} - 2\,n^{3} x^{2} - 2\,n^{3} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
111
2549
32136966
48422412616450
533012001080320252955
612875940742536006753618963
750052802845045308009625126049119812

All 15 rows

Technical details
Node id
n-731ce0c1f40f7072 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-f0acff1b7583ce22

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:     0;
  1:     1;
  2:     5,      4;
  3:    21,     36,       9;
  4:    84,    224,     126,      16;
  5:   330,   1200,    1080,     320,     25;
  6:  1287,   5940,    7425,    3600,    675,     36;
  7:  5005,  28028,   45045,   30800,   9625,   1260,    49;
  8: 19448, 128128,  252252,  224224, 100100,  22176,  2156,   64;
  9: 75582, 572832, 1336608, 1467648, 859950, 275184, 45864, 3456, 81;
%F Formula
The row polynomials satisfy (2*n^3+3*n^2-3*n-2+(n^4-3*n^2+2*n)*x)*P(n,x) = (8*n^3+8*n^2+2*n+(4*n^4+10*n^3-10*n-4)*x+(3*n^4-4*n^2-n+2)*x^2)*P(n-1,x) - ((4*n^2+4*n+1)*x+(2*n^3+6*n^2+4*n)*x^2+(2*n^3+2*n^2-1)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^3 + 8*n^2 + 2*n + (4*n^4 + 10*n^3 - 10*n - 4)*x + (3*n^4 - 4*n^2 - n + 2)*x^2)*P[n - 1, x] - ((4*n^2 + 4*n + 1)*x + (2*n^3 + 6*n^2 + 4*n)*x^2 + (2*n^3 + 2*n^2 - 1)*x^3)*D[P[n - 1, x], x])/(2*n^3 + 3*n^2 - 3*n - 2 + (n^4 - 3*n^2 + 2*n)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((8*n^3 + 8*n^2 + 2*n + (4*n^4 + 10*n^3 - 10*n - 4)*x + (3*n^4 - 4*n^2 - n + 2)*x^2)*v[i - 1] - ((4*n^2 + 4*n + 1)*x + (2*n^3 + 6*n^2 + 4*n)*x^2 + (2*n^3 + 2*n^2 - 1)*x^3)*deriv(v[i - 1], x))/(2*n^3 + 3*n^2 - 3*n - 2 + (n^4 - 3*n^2 + 2*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

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