Real-rooted atlas

A055151 → D → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-4 - 40\,x - 16\,x^{2} + 40\,n x + 48\,n x^{2} + n^{2} + 2\,n^{2} x - 48\,n^{2} x^{2} - 10\,n^{3} x + 12\,n^{3} x^{2} + 2\,n^{4} x + 9\,n^{4} x^{2} - 6\,n^{5} x^{2} + n^{6} x^{2}\right)\,P_{n}(x) = \left(2\,n - 28\,n x - 40\,n x^{2} + n^{2} + 6\,n^{2} x + 128\,n^{2} x^{2} - 6\,n^{3} x - 66\,n^{3} x^{2} - 12\,n^{3} x^{3} + 2\,n^{4} x + 13\,n^{4} x^{2} + 28\,n^{4} x^{3} - 4\,n^{5} x^{2} - 20\,n^{5} x^{3} + n^{6} x^{2} + 4\,n^{6} x^{3}\right)\,P_{n-1}(x) + \left(2\,n x - 4\,n x^{2} - 16\,n x^{3} - 12\,n^{2} x^{2} + 48\,n^{2} x^{3} + 4\,n^{3} x^{2} - 14\,n^{3} x^{3} - 8\,n^{3} x^{4} - 4\,n^{4} x^{3} + 16\,n^{4} x^{4} + 2\,n^{5} x^{3} - 8\,n^{5} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
10
211
333
461622
5108090
615240135390
7215609451526
828112037808965824
936201611340806421456
104533602835040320525077325
115552806237014784057750273295
1266792012474044352034650028512951258
137811440231660115315215015003706563268486
14911601640540526906885255250259459214714711109189
Technical details
Node id
n-d26385759ec0bf4b · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-f6306ce70d855d19

Observed on the stored rows ?

  • Degrees: -1, -1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6 (half 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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:  0;
  1:  0;
  2:  1;
  3:  3;
  4:  6,   16;
  5: 10,   80;
  6: 15,  240,   135;
  7: 21,  560,   945;
  8: 28, 1120,  3780,  896;
  9: 36, 2016, 11340, 8064;
%F Formula
The row polynomials satisfy (n^2-4+(2*n^4-10*n^3+2*n^2+40*n-40)*x+(n^6-6*n^5+9*n^4+12*n^3-48*n^2+48*n-16)*x^2)*P(n,x) = (n^2+2*n+(2*n^4-6*n^3+6*n^2-28*n)*x+(n^6-4*n^5+13*n^4-66*n^3+128*n^2-40*n)*x^2+(4*n^6-20*n^5+28*n^4-12*n^3)*x^3)*P(n-1,x) + (2*n*x+(4*n^3-12*n^2-4*n)*x^2+(2*n^5-4*n^4-14*n^3+48*n^2-16*n)*x^3-(8*n^5-16*n^4+8*n^3)*x^4)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + 2*n + (2*n^4 - 6*n^3 + 6*n^2 - 28*n)*x + (n^6 - 4*n^5 + 13*n^4 - 66*n^3 + 128*n^2 - 40*n)*x^2 + (4*n^6 - 20*n^5 + 28*n^4 - 12*n^3)*x^3)*P[n - 1, x] + (2*n*x + (4*n^3 - 12*n^2 - 4*n)*x^2 + (2*n^5 - 4*n^4 - 14*n^3 + 48*n^2 - 16*n)*x^3 - (8*n^5 - 16*n^4 + 8*n^3)*x^4)*D[P[n - 1, x], x])/(n^2 - 4 + (2*n^4 - 10*n^3 + 2*n^2 + 40*n - 40)*x + (n^6 - 6*n^5 + 9*n^4 + 12*n^3 - 48*n^2 + 48*n - 16)*x^2)]];
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] = 0; v[3] = 1; for(n = 3, N, my(i = n + 1); v[i] = ((n^2 + 2*n + (2*n^4 - 6*n^3 + 6*n^2 - 28*n)*x + (n^6 - 4*n^5 + 13*n^4 - 66*n^3 + 128*n^2 - 40*n)*x^2 + (4*n^6 - 20*n^5 + 28*n^4 - 12*n^3)*x^3)*v[i - 1] + (2*n*x + (4*n^3 - 12*n^2 - 4*n)*x^2 + (2*n^5 - 4*n^4 - 14*n^3 + 48*n^2 - 16*n)*x^3 - (8*n^5 - 16*n^4 + 8*n^3)*x^4)*deriv(v[i - 1], x))/(n^2 - 4 + (2*n^4 - 10*n^3 + 2*n^2 + 40*n - 40)*x + (n^6 - 6*n^5 + 9*n^4 + 12*n^3 - 48*n^2 + 48*n - 16)*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