Real-rooted atlas

A208509 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(8 + 28\,x + 12\,x^{2} - 16\,n + 8\,n x - 2\,n x^{2} - 52\,n^{2} x - 26\,n^{2} x^{2} + n^{2} x^{3} + 40\,n^{3} x - 6\,n^{3} x^{2} + 3\,n^{3} x^{3} + 2\,n^{4} x^{2} + 3\,n^{4} x^{3} - 4\,n^{5} x^{2} + n^{5} x^{3}\right)\,P_{n}(x) = \left(-8 + 132\,x + 140\,x^{2} + 28\,x^{3} - 4\,x^{4} - 16\,n - 88\,n x - 194\,n x^{2} + 6\,n x^{3} - 8\,n x^{4} - 12\,n^{2} x - 78\,n^{2} x^{2} - 3\,n^{2} x^{3} - n^{2} x^{4} + 40\,n^{3} x + 66\,n^{3} x^{2} + n^{3} x^{3} + 7\,n^{3} x^{4} - 2\,n^{4} x^{2} - 5\,n^{4} x^{3} + 5\,n^{4} x^{4} - 4\,n^{5} x^{2} - 3\,n^{5} x^{3} + n^{5} x^{4}\right)\,P_{n-1}(x) + \left(-32\,x - 8\,x^{2} + 8\,x^{3} + 40\,x^{4} - 8\,x^{5} + 96\,n x^{2} - 148\,n x^{3} + 76\,n x^{4} - 24\,n x^{5} + 80\,n^{2} x^{2} - 132\,n^{2} x^{3} + 78\,n^{2} x^{4} - 26\,n^{2} x^{5} - 32\,n^{3} x^{3} + 44\,n^{3} x^{4} - 12\,n^{3} x^{5} - 8\,n^{4} x^{3} + 10\,n^{4} x^{4} - 2\,n^{4} x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
011
133
2549
372027
4956974
51112063194
61322024316492
7153646931441216

All 15 rows

Technical details
Node id
n-8215d84dfeeafecb · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-0eb54671b257fa23

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half affine: slope 1, intercept 0)
  • 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:  3;
  2:  5,   4;
  3:  7,  20;
  4:  9,  56,    9;
  5: 11, 120,   63;
  6: 13, 220,  243,   16;
  7: 15, 364,  693,  144;
  8: 17, 560, 1638,  704,  25;
  9: 19, 816, 3402, 2496, 275;
%F Formula
The row polynomials satisfy -(16*n-8-(40*n^3-52*n^2+8*n+28)*x+(4*n^5-2*n^4+6*n^3+26*n^2+2*n-12)*x^2-(n^5+3*n^4+3*n^3+n^2)*x^3)*P(n,x) = -(16*n+8-(40*n^3-12*n^2-88*n+132)*x+(4*n^5+2*n^4-66*n^3+78*n^2+194*n-140)*x^2+(3*n^5+5*n^4-n^3+3*n^2-6*n-28)*x^3-(n^5+5*n^4+7*n^3-n^2-8*n-4)*x^4)*P(n-1,x) - (32*x-(80*n^2+96*n-8)*x^2+(8*n^4+32*n^3+132*n^2+148*n-8)*x^3-(10*n^4+44*n^3+78*n^2+76*n+40)*x^4+(2*n^4+12*n^3+26*n^2+24*n+8)*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(16*n + 8 - (40*n^3 - 12*n^2 - 88*n + 132)*x + (4*n^5 + 2*n^4 - 66*n^3 + 78*n^2 + 194*n - 140)*x^2 + (3*n^5 + 5*n^4 - n^3 + 3*n^2 - 6*n - 28)*x^3 - (n^5 + 5*n^4 + 7*n^3 - n^2 - 8*n - 4)*x^4)*P[n - 1, x] - (32*x - (80*n^2 + 96*n - 8)*x^2 + (8*n^4 + 32*n^3 + 132*n^2 + 148*n - 8)*x^3 - (10*n^4 + 44*n^3 + 78*n^2 + 76*n + 40)*x^4 + (2*n^4 + 12*n^3 + 26*n^2 + 24*n + 8)*x^5)*D[P[n - 1, x], x])/(-16*n + 8 + (40*n^3 - 52*n^2 + 8*n + 28)*x - (4*n^5 - 2*n^4 + 6*n^3 + 26*n^2 + 2*n - 12)*x^2 + (n^5 + 3*n^4 + 3*n^3 + n^2)*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(16*n + 8 - (40*n^3 - 12*n^2 - 88*n + 132)*x + (4*n^5 + 2*n^4 - 66*n^3 + 78*n^2 + 194*n - 140)*x^2 + (3*n^5 + 5*n^4 - n^3 + 3*n^2 - 6*n - 28)*x^3 - (n^5 + 5*n^4 + 7*n^3 - n^2 - 8*n - 4)*x^4)*v[i - 1] - (32*x - (80*n^2 + 96*n - 8)*x^2 + (8*n^4 + 32*n^3 + 132*n^2 + 148*n - 8)*x^3 - (10*n^4 + 44*n^3 + 78*n^2 + 76*n + 40)*x^4 + (2*n^4 + 12*n^3 + 26*n^2 + 24*n + 8)*x^5)*deriv(v[i - 1], x))/(-16*n + 8 + (40*n^3 - 52*n^2 + 8*n + 28)*x - (4*n^5 - 2*n^4 + 6*n^3 + 26*n^2 + 2*n - 12)*x^2 + (n^5 + 3*n^4 + 3*n^3 + n^2)*x^3)); 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. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.