Real-rooted atlas

A078812 → D → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-16 + 16\,x + 8\,n + 8\,n x - 16\,n^{2} x - 16\,n^{2} x^{2} + 4\,n^{3} x + 24\,n^{3} x^{2} - 12\,n^{4} x^{2} + 2\,n^{5} x^{2}\right)\,P_{n}(x) = \left(8 + 8\,x - 4\,x^{2} + 8\,n - 6\,n x^{2} - 4\,n^{2} x + 8\,n^{2} x^{2} + n^{2} x^{3} + 4\,n^{3} x + 2\,n^{3} x^{2} - n^{3} x^{3} - 6\,n^{4} x^{2} - n^{4} x^{3} + 2\,n^{5} x^{2} + n^{5} x^{3}\right)\,P_{n-1}(x) + \left(16\,x - 4\,x^{2} - 2\,x^{3} - 8\,n x^{2} - 2\,n x^{3} + 8\,n^{2} x^{2} + 6\,n^{2} x^{3} + n^{2} x^{4} - 8\,n^{3} x^{3} - 2\,n^{3} x^{4} + 4\,n^{4} x^{3} + n^{4} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
10
211
34812
410482785
52016821664468
6354489726401252220
75610083240352015002169540
88420168910140809750302434338207

All 15 rows

Technical details
Node id
n-83da9a53534aacc2 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-14e0c58803402a0f

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 -2)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:   0;
   2:   1;
   3:   4,    8;
   4:  10,   48,    27;
   5:  20,  168,   216,     64;
   6:  35,  448,   972,    640,    125;
   7:  56, 1008,  3240,   3520,   1500,    216;
   8:  84, 2016,  8910,  14080,   9750,   3024,   343;
   9: 120, 3696, 21384,  45760,  45500,  22680,  5488,  512;
  10: 165, 6336, 46332, 128128, 170625, 120960, 46648, 9216, 729;
%F Formula
The row polynomials satisfy (8*n-16+(4*n^3-16*n^2+8*n+16)*x+(2*n^5-12*n^4+24*n^3-16*n^2)*x^2)*P(n,x) = (8*n+8+(4*n^3-4*n^2+8)*x+(2*n^5-6*n^4+2*n^3+8*n^2-6*n-4)*x^2+(n^5-n^4-n^3+n^2)*x^3)*P(n-1,x) + (16*x+(8*n^2-8*n-4)*x^2+(4*n^4-8*n^3+6*n^2-2*n-2)*x^3+(n^4-2*n^3+n^2)*x^4)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((8*n + 8 + (4*n^3 - 4*n^2 + 8)*x + (2*n^5 - 6*n^4 + 2*n^3 + 8*n^2 - 6*n - 4)*x^2 + (n^5 - n^4 - n^3 + n^2)*x^3)*P[n - 1, x] + (16*x + (8*n^2 - 8*n - 4)*x^2 + (4*n^4 - 8*n^3 + 6*n^2 - 2*n - 2)*x^3 + (n^4 - 2*n^3 + n^2)*x^4)*D[P[n - 1, x], x])/(8*n - 16 + (4*n^3 - 16*n^2 + 8*n + 16)*x + (2*n^5 - 12*n^4 + 24*n^3 - 16*n^2)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = ((8*n + 8 + (4*n^3 - 4*n^2 + 8)*x + (2*n^5 - 6*n^4 + 2*n^3 + 8*n^2 - 6*n - 4)*x^2 + (n^5 - n^4 - n^3 + n^2)*x^3)*v[i - 1] + (16*x + (8*n^2 - 8*n - 4)*x^2 + (4*n^4 - 8*n^3 + 6*n^2 - 2*n - 2)*x^3 + (n^4 - 2*n^3 + n^2)*x^4)*deriv(v[i - 1], x))/(8*n - 16 + (4*n^3 - 16*n^2 + 8*n + 16)*x + (2*n^5 - 12*n^4 + 24*n^3 - 16*n^2)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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