Real-rooted atlas

A008459 → D → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 - x - 3\,n + 5\,n x + 3\,n^{2} - 10\,n^{2} x - n^{3} + 10\,n^{3} x - 5\,n^{4} x + n^{5} x\right)\,P_{n}(x) = \left(n^{2} + 9\,n^{2} x + 2\,n^{2} x^{2} - n^{3} - 3\,n^{3} x + 2\,n^{3} x^{2} - 3\,n^{4} x - 7\,n^{4} x^{2} + n^{5} x + 3\,n^{5} x^{2}\right)\,P_{n-1}(x) + \left(-2\,n^{2} x + n^{2} x^{2} + n^{2} x^{3} - n^{3} x^{2} + n^{3} x^{3} + 2\,n^{4} x^{2} - 2\,n^{4} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
111
24812
397227108
41628843264800
525800270016001255250
63618001080014400450021631752
749352833075784005512510584343181104

All 15 rows

Technical details
Node id
n-615dcc41243fb001 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-160840ed44c00ba4

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:  4,     8;
  3:  9,    72,     27;
  4: 16,   288,    432,      64;
  5: 25,   800,   2700,    1600,     125;
  6: 36,  1800,  10800,   14400,    4500,     216;
  7: 49,  3528,  33075,   78400,   55125,   10584,    343;
  8: 64,  6272,  84672,  313600,  392000,  169344,  21952,   512;
  9: 81, 10368, 190512, 1016064, 1984500, 1524096, 444528, 41472, 729;
%F Formula
The row polynomials satisfy -(n^3-3*n^2+3*n-1-(n^5-5*n^4+10*n^3-10*n^2+5*n-1)*x)*P(n,x) = -(n^3-n^2-(n^5-3*n^4-3*n^3+9*n^2)*x-(3*n^5-7*n^4+2*n^3+2*n^2)*x^2)*P(n-1,x) - (2*n^2*x-(2*n^4-n^3+n^2)*x^2+(2*n^4-n^3-n^2)*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[(-(n^3 - n^2 - (n^5 - 3*n^4 - 3*n^3 + 9*n^2)*x - (3*n^5 - 7*n^4 + 2*n^3 + 2*n^2)*x^2)*P[n - 1, x] - (2*n^2*x - (2*n^4 - n^3 + n^2)*x^2 + (2*n^4 - n^3 - n^2)*x^3)*D[P[n - 1, x], x])/(-n^3 + 3*n^2 - 3*n + 1 + (n^5 - 5*n^4 + 10*n^3 - 10*n^2 + 5*n - 1)*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] = (-(n^3 - n^2 - (n^5 - 3*n^4 - 3*n^3 + 9*n^2)*x - (3*n^5 - 7*n^4 + 2*n^3 + 2*n^2)*x^2)*v[i - 1] - (2*n^2*x - (2*n^4 - n^3 + n^2)*x^2 + (2*n^4 - n^3 - n^2)*x^3)*deriv(v[i - 1], x))/(-n^3 + 3*n^2 - 3*n + 1 + (n^5 - 5*n^4 + 10*n^3 - 10*n^2 + 5*n - 1)*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