Real-rooted atlas

A367270 → x ↦ x+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-n + n x + 5\,n^{2} - 7\,n^{2} x - 8\,n^{3} + 18\,n^{3} x - n^{3} x^{2} + 4\,n^{4} - 18\,n^{4} x + 3\,n^{4} x^{2} + 6\,n^{5} x - 3\,n^{5} x^{2} + n^{6} x^{2}\right)\,P_{n}(x) = \left(2 + 6\,x + 6\,x^{2} + 2\,x^{3} - 14\,n - 34\,n x - 27\,n x^{2} - 5\,n x^{3} + 36\,n^{2} + 60\,n^{2} x + 22\,n^{2} x^{2} - 40\,n^{3} - 20\,n^{3} x + 34\,n^{3} x^{2} + 9\,n^{3} x^{3} + 16\,n^{4} - 36\,n^{4} x - 55\,n^{4} x^{2} - 6\,n^{4} x^{3} + 24\,n^{5} x + 16\,n^{5} x^{2} - 3\,n^{5} x^{3} + 4\,n^{6} x^{2} + 3\,n^{6} x^{3}\right)\,P_{n-1}(x) + \left(x + 3\,x^{2} + 3\,x^{3} + x^{4} - 6\,n x - 14\,n x^{2} - 10\,n x^{3} - 2\,n x^{4} + 12\,n^{2} x + 16\,n^{2} x^{2} + 3\,n^{2} x^{3} - n^{2} x^{4} - 8\,n^{3} x + 4\,n^{3} x^{2} + 16\,n^{3} x^{3} + 4\,n^{3} x^{4} - 12\,n^{4} x^{2} - 13\,n^{4} x^{3} - n^{4} x^{4} - 2\,n^{5} x^{3} - 2\,n^{5} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
011
111
23811
310482785
43524027064609
5126112018909601254221
64625040113408960262521628643
71716221766237067200315006048343191353

All 15 rows

Technical details
Node id
n-f4dba2b3aa45027c · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-548f1af0501a83b7

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
  • 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:     1;
  2:     3,      8;
  3:    10,     48,      27;
  4:    35,    240,     270,      64;
  5:   126,   1120,    1890,     960,     125;
  6:   462,   5040,   11340,    8960,    2625,    216;
  7:  1716,  22176,   62370,   67200,   31500,   6048,    343;
  8:  6435,  96096,  324324,  443520,  288750,  90720,  12348,   512;
  9: 24310, 411840, 1621620, 2690688, 2252250, 997920, 226380, 23040, 729;
%F Formula
The row polynomials satisfy (4*n^4-8*n^3+5*n^2-n+(6*n^5-18*n^4+18*n^3-7*n^2+n)*x+(n^6-3*n^5+3*n^4-n^3)*x^2)*P(n,x) = (16*n^4-40*n^3+36*n^2-14*n+2+(24*n^5-36*n^4-20*n^3+60*n^2-34*n+6)*x+(4*n^6+16*n^5-55*n^4+34*n^3+22*n^2-27*n+6)*x^2+(3*n^6-3*n^5-6*n^4+9*n^3-5*n+2)*x^3)*P(n-1,x) - ((8*n^3-12*n^2+6*n-1)*x+(12*n^4-4*n^3-16*n^2+14*n-3)*x^2+(2*n^5+13*n^4-16*n^3-3*n^2+10*n-3)*x^3+(2*n^5+n^4-4*n^3+n^2+2*n-1)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((16*n^4 - 40*n^3 + 36*n^2 - 14*n + 2 + (24*n^5 - 36*n^4 - 20*n^3 + 60*n^2 - 34*n + 6)*x + (4*n^6 + 16*n^5 - 55*n^4 + 34*n^3 + 22*n^2 - 27*n + 6)*x^2 + (3*n^6 - 3*n^5 - 6*n^4 + 9*n^3 - 5*n + 2)*x^3)*P[n - 1, x] - ((8*n^3 - 12*n^2 + 6*n - 1)*x + (12*n^4 - 4*n^3 - 16*n^2 + 14*n - 3)*x^2 + (2*n^5 + 13*n^4 - 16*n^3 - 3*n^2 + 10*n - 3)*x^3 + (2*n^5 + n^4 - 4*n^3 + n^2 + 2*n - 1)*x^4)*D[P[n - 1, x], x])/(4*n^4 - 8*n^3 + 5*n^2 - n + (6*n^5 - 18*n^4 + 18*n^3 - 7*n^2 + n)*x + (n^6 - 3*n^5 + 3*n^4 - n^3)*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; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((16*n^4 - 40*n^3 + 36*n^2 - 14*n + 2 + (24*n^5 - 36*n^4 - 20*n^3 + 60*n^2 - 34*n + 6)*x + (4*n^6 + 16*n^5 - 55*n^4 + 34*n^3 + 22*n^2 - 27*n + 6)*x^2 + (3*n^6 - 3*n^5 - 6*n^4 + 9*n^3 - 5*n + 2)*x^3)*v[i - 1] - ((8*n^3 - 12*n^2 + 6*n - 1)*x + (12*n^4 - 4*n^3 - 16*n^2 + 14*n - 3)*x^2 + (2*n^5 + 13*n^4 - 16*n^3 - 3*n^2 + 10*n - 3)*x^3 + (2*n^5 + n^4 - 4*n^3 + n^2 + 2*n - 1)*x^4)*deriv(v[i - 1], x))/(4*n^4 - 8*n^3 + 5*n^2 - n + (6*n^5 - 18*n^4 + 18*n^3 - 7*n^2 + n)*x + (n^6 - 3*n^5 + 3*n^4 - n^3)*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