Real-rooted atlas

A062196 → D → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-1 - x + 2\,n^{2} + 3\,n^{2} x - n^{4} - 3\,n^{4} x + n^{6} x\right)\,P_{n}(x) = \left(2\,n - 2\,n x - 4\,n x^{2} + n^{2} + 7\,n^{2} x + 10\,n^{2} x^{2} - 2\,n^{3} - 12\,n^{3} x - 4\,n^{3} x^{2} - n^{4} - 8\,n^{4} x - n^{4} x^{2} + 2\,n^{5} x + 8\,n^{5} x^{2} + n^{6} x + 3\,n^{6} x^{2}\right)\,P_{n-1}(x) + \left(2\,n x^{2} - 2\,n x^{3} - 4\,n^{2} x + n^{2} x^{2} + 3\,n^{2} x^{3} - 2\,n^{3} x + 6\,n^{3} x^{2} - 4\,n^{3} x^{3} + 7\,n^{4} x^{2} - 7\,n^{4} x^{3} + 2\,n^{5} x^{2} - 2\,n^{5} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
133
284856
315240270525
42472021609603864
5351680945011200262524990
64833603024067200420006048148896
76360487938028224033075012700812348837837

All 15 rows

Technical details
Node id
n-da3963458b44f2af · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-597cc2b5aadef49f

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 -1)
  • 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:  3;
  2:  8,    48;
  3: 15,   240,    270;
  4: 24,   720,   2160,     960;
  5: 35,  1680,   9450,   11200,    2625;
  6: 48,  3360,  30240,   67200,   42000,    6048;
  7: 63,  6048,  79380,  282240,  330750,  127008,   12348;
  8: 80, 10080, 181440,  940800, 1764000, 1270080,  329280,  23040;
  9: 99, 15840, 374220, 2661120, 7276500, 8382528, 4074840, 760320, 40095;
%F Formula
The row polynomials satisfy -(n^4-2*n^2+1-(n^6-3*n^4+3*n^2-1)*x)*P(n,x) = -(n^4+2*n^3-n^2-2*n-(n^6+2*n^5-8*n^4-12*n^3+7*n^2-2*n)*x-(3*n^6+8*n^5-n^4-4*n^3+10*n^2-4*n)*x^2)*P(n-1,x) - ((2*n^3+4*n^2)*x-(2*n^5+7*n^4+6*n^3+n^2+2*n)*x^2+(2*n^5+7*n^4+4*n^3-3*n^2+2*n)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^4 + 2*n^3 - n^2 - 2*n - (n^6 + 2*n^5 - 8*n^4 - 12*n^3 + 7*n^2 - 2*n)*x - (3*n^6 + 8*n^5 - n^4 - 4*n^3 + 10*n^2 - 4*n)*x^2)*P[n - 1, x] - ((2*n^3 + 4*n^2)*x - (2*n^5 + 7*n^4 + 6*n^3 + n^2 + 2*n)*x^2 + (2*n^5 + 7*n^4 + 4*n^3 - 3*n^2 + 2*n)*x^3)*D[P[n - 1, x], x])/(-n^4 + 2*n^2 - 1 + (n^6 - 3*n^4 + 3*n^2 - 1)*x)]];
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] = 3; for(n = 2, N, my(i = n + 1); v[i] = (-(n^4 + 2*n^3 - n^2 - 2*n - (n^6 + 2*n^5 - 8*n^4 - 12*n^3 + 7*n^2 - 2*n)*x - (3*n^6 + 8*n^5 - n^4 - 4*n^3 + 10*n^2 - 4*n)*x^2)*v[i - 1] - ((2*n^3 + 4*n^2)*x - (2*n^5 + 7*n^4 + 6*n^3 + n^2 + 2*n)*x^2 + (2*n^5 + 7*n^4 + 4*n^3 - 3*n^2 + 2*n)*x^3)*deriv(v[i - 1], x))/(-n^4 + 2*n^2 - 1 + (n^6 - 3*n^4 + 3*n^2 - 1)*x)); 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.