Real-rooted atlas

A280580 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(6\,n - 12\,n^{2} + n^{2} x - 3\,n^{3} x + 2\,n^{4} x\right)\,P_{n}(x) = \left(-36 - 72\,x - 6\,x^{2} + 96\,n + 138\,n x + 13\,n x^{2} - 48\,n^{2} - 50\,n^{2} x - 3\,n^{2} x^{2} - 24\,n^{3} x - 10\,n^{3} x^{2} + 8\,n^{4} x + 6\,n^{4} x^{2}\right)\,P_{n-1}(x) + \left(72\,x - 6\,x^{2} - 3\,x^{3} - 96\,n x + 4\,n x^{2} + 5\,n x^{3} - 4\,n^{2} x^{2} + n^{2} x^{3} + 16\,n^{3} x^{2} - 4\,n^{3} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2123
3212317
4560454114
5142804201125831
64212603150168022566363
7132554420790184804950396750299
84292402412612616816875075120126378406479

All 15 rows

Technical details
Node id
n-d0d869eaaa277eca · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-4f57bdcd9ba55419

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:    1;
   2:    1,      2;
   3:    2,     12,       3;
   4:    5,     60,      45,       4;
   5:   14,    280,     420,     112,       5;
   6:   42,   1260,    3150,    1680,     225,       6;
   7:  132,   5544,   20790,   18480,    4950,     396,      7;
   8:  429,  24024,  126126,  168168,   75075,   12012,    637,     8;
   9: 1430, 102960,  720720, 1345344,  900900,  240240,  25480,   960,    9;
  10: 4862, 437580, 3938220, 9801792, 9189180, 3675672, 649740, 48960, 1377, 10;
%F Formula
The row polynomials satisfy -(12*n^2-6*n-(2*n^4-3*n^3+n^2)*x)*P(n,x) = -(48*n^2-96*n+36-(8*n^4-24*n^3-50*n^2+138*n-72)*x-(6*n^4-10*n^3-3*n^2+13*n-6)*x^2)*P(n-1,x) - ((96*n-72)*x-(16*n^3-4*n^2+4*n-6)*x^2+(4*n^3-n^2-5*n+3)*x^3)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(48*n^2 - 96*n + 36 - (8*n^4 - 24*n^3 - 50*n^2 + 138*n - 72)*x - (6*n^4 - 10*n^3 - 3*n^2 + 13*n - 6)*x^2)*P[n - 1, x] - ((96*n - 72)*x - (16*n^3 - 4*n^2 + 4*n - 6)*x^2 + (4*n^3 - n^2 - 5*n + 3)*x^3)*D[P[n - 1, x], x])/(-12*n^2 + 6*n + (2*n^4 - 3*n^3 + n^2)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(48*n^2 - 96*n + 36 - (8*n^4 - 24*n^3 - 50*n^2 + 138*n - 72)*x - (6*n^4 - 10*n^3 - 3*n^2 + 13*n - 6)*x^2)*v[i - 1] - ((96*n - 72)*x - (16*n^3 - 4*n^2 + 4*n - 6)*x^2 + (4*n^3 - n^2 - 5*n + 3)*x^3)*deriv(v[i - 1], x))/(-12*n^2 + 6*n + (2*n^4 - 3*n^3 + n^2)*x)); 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. [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.