Real-rooted atlas

A034807 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(n x - n^{2} - n^{2} x - n^{2} x^{2} + n^{3} - 3\,n^{3} x - n^{3} x^{2} + 3\,n^{4} x + n^{4} x^{2} + n^{5} x^{2}\right)\,P_{n}(x) = \left(-8\,x^{2} - 8\,x^{3} + n x - 16\,n x^{3} - n^{2} - 9\,n^{2} x - 11\,n^{2} x^{2} - 2\,n^{2} x^{3} + n^{3} + 5\,n^{3} x + 7\,n^{3} x^{2} + 14\,n^{3} x^{3} + 3\,n^{4} x + 11\,n^{4} x^{2} + 10\,n^{4} x^{3} + n^{5} x^{2} + 2\,n^{5} x^{3}\right)\,P_{n-1}(x) + \left(-2\,x^{2} - 12\,x^{3} - 16\,x^{4} - 3\,n x^{2} - 24\,n x^{3} - 48\,n x^{4} - n^{2} x - 10\,n^{2} x^{2} - 37\,n^{2} x^{3} - 52\,n^{2} x^{4} - 3\,n^{3} x^{2} - 18\,n^{3} x^{3} - 24\,n^{3} x^{4} - n^{4} x^{3} - 4\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 2\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
022
111
2189
311213
41161835
51204566
61248132138
7128126112267

All 15 rows

Technical details
Node id
n-c9d99dccbf962b96 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-c5ae5c76bbcc6daf

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: 2;
  1: 1;
  2: 1,  8;
  3: 1, 12;
  4: 1, 16,  18;
  5: 1, 20,  45;
  6: 1, 24,  81,  32;
  7: 1, 28, 126, 112;
  8: 1, 32, 180, 256,  50;
  9: 1, 36, 243, 480, 225;
%F Formula
The row polynomials satisfy (n^3-n^2+(3*n^4-3*n^3-n^2+n)*x+(n^5+n^4-n^3-n^2)*x^2)*P(n,x) = (n^3-n^2+(3*n^4+5*n^3-9*n^2+n)*x+(n^5+11*n^4+7*n^3-11*n^2-8)*x^2+(2*n^5+10*n^4+14*n^3-2*n^2-16*n-8)*x^3)*P(n-1,x) - (n^2*x+(3*n^3+10*n^2+3*n+2)*x^2+(n^4+18*n^3+37*n^2+24*n+12)*x^3+(4*n^4+24*n^3+52*n^2+48*n+16)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 2, P(1,x) = 1.
%t Mathematica
P[0, x_] := 2;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^3 - n^2 + (3*n^4 + 5*n^3 - 9*n^2 + n)*x + (n^5 + 11*n^4 + 7*n^3 - 11*n^2 - 8)*x^2 + (2*n^5 + 10*n^4 + 14*n^3 - 2*n^2 - 16*n - 8)*x^3)*P[n - 1, x] - (n^2*x + (3*n^3 + 10*n^2 + 3*n + 2)*x^2 + (n^4 + 18*n^3 + 37*n^2 + 24*n + 12)*x^3 + (4*n^4 + 24*n^3 + 52*n^2 + 48*n + 16)*x^4)*D[P[n - 1, x], x])/(n^3 - n^2 + (3*n^4 - 3*n^3 - n^2 + n)*x + (n^5 + n^4 - n^3 - n^2)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((n^3 - n^2 + (3*n^4 + 5*n^3 - 9*n^2 + n)*x + (n^5 + 11*n^4 + 7*n^3 - 11*n^2 - 8)*x^2 + (2*n^5 + 10*n^4 + 14*n^3 - 2*n^2 - 16*n - 8)*x^3)*v[i - 1] - (n^2*x + (3*n^3 + 10*n^2 + 3*n + 2)*x^2 + (n^4 + 18*n^3 + 37*n^2 + 24*n + 12)*x^3 + (4*n^4 + 24*n^3 + 52*n^2 + 48*n + 16)*x^4)*deriv(v[i - 1], x))/(n^3 - n^2 + (3*n^4 - 3*n^3 - n^2 + n)*x + (n^5 + n^4 - n^3 - 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 - 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. [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.