Real-rooted atlas

A280580 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-4\,n + 8\,n x - n x^{2} + 11\,n^{2} - 18\,n^{2} x + 2\,n^{2} x^{2} - 7\,n^{3} + 4\,n^{3} x + 2\,n^{4}\right)\,P_{n}(x) = \left(12 - 60\,x - 36\,x^{2} + 6\,x^{3} - 38\,n + 148\,n x + 80\,n x^{2} - 12\,n x^{3} + 38\,n^{2} - 72\,n^{2} x - 58\,n^{2} x^{2} + 6\,n^{2} x^{3} - 24\,n^{3} - 10\,n^{3} x + 12\,n^{3} x^{2} + 8\,n^{4} + 6\,n^{4} x\right)\,P_{n-1}(x) + \left(24\,x + 66\,x^{2} - 30\,x^{3} + 3\,x^{4} - 12\,n x - 117\,n x^{2} + 46\,n x^{3} - 4\,n x^{4} - 36\,n^{2} x + 41\,n^{2} x^{2} - 8\,n^{2} x^{3} + 16\,n^{3} x - 4\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2213
388117
43560181114
5154420224321831
6672273023106005016363
7290416632207908580132072150299
81244196096168168102102250252548981406479

All 15 rows

Technical details
Node id
n-c3958ba407bf5cdb · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-812455d3af977983

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:      2,       1;
   3:      8,       8,       1;
   4:     35,      60,      18,       1;
   5:    154,     420,     224,      32,       1;
   6:    672,    2730,    2310,     600,      50,       1;
   7:   2904,   16632,   20790,    8580,    1320,      72,      1;
   8:  12441,   96096,  168168,  102102,   25025,    2548,     98,    1;
   9:  52910,  531960, 1249248, 1057056,  380380,   61880,   4480,  128,   1;
  10: 223652, 2844270, 8664084, 9801792, 4900896, 1169532, 135660, 7344, 162, 1;
%F Formula
The row polynomials satisfy (2*n^4-7*n^3+11*n^2-4*n+(4*n^3-18*n^2+8*n)*x+(2*n^2-n)*x^2)*P(n,x) = (8*n^4-24*n^3+38*n^2-38*n+12+(6*n^4-10*n^3-72*n^2+148*n-60)*x+(12*n^3-58*n^2+80*n-36)*x^2+(6*n^2-12*n+6)*x^3)*P(n-1,x) + ((16*n^3-36*n^2-12*n+24)*x-(4*n^3-41*n^2+117*n-66)*x^2-(8*n^2-46*n+30)*x^3-(4*n-3)*x^4)*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[((8*n^4 - 24*n^3 + 38*n^2 - 38*n + 12 + (6*n^4 - 10*n^3 - 72*n^2 + 148*n - 60)*x + (12*n^3 - 58*n^2 + 80*n - 36)*x^2 + (6*n^2 - 12*n + 6)*x^3)*P[n - 1, x] + ((16*n^3 - 36*n^2 - 12*n + 24)*x - (4*n^3 - 41*n^2 + 117*n - 66)*x^2 - (8*n^2 - 46*n + 30)*x^3 - (4*n - 3)*x^4)*D[P[n - 1, x], x])/(2*n^4 - 7*n^3 + 11*n^2 - 4*n + (4*n^3 - 18*n^2 + 8*n)*x + (2*n^2 - n)*x^2)]];
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] = ((8*n^4 - 24*n^3 + 38*n^2 - 38*n + 12 + (6*n^4 - 10*n^3 - 72*n^2 + 148*n - 60)*x + (12*n^3 - 58*n^2 + 80*n - 36)*x^2 + (6*n^2 - 12*n + 6)*x^3)*v[i - 1] + ((16*n^3 - 36*n^2 - 12*n + 24)*x - (4*n^3 - 41*n^2 + 117*n - 66)*x^2 - (8*n^2 - 46*n + 30)*x^3 - (4*n - 3)*x^4)*deriv(v[i - 1], x))/(2*n^4 - 7*n^3 + 11*n^2 - 4*n + (4*n^3 - 18*n^2 + 8*n)*x + (2*n^2 - n)*x^2)); 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.