Real-rooted atlas

A280580 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(-n - 2\,n x - n x^{2} + 2\,n^{2} + 4\,n^{2} x + 2\,n^{2} x^{2}\right)\,P_{n}(x) = \left(6 - 36\,x^{2} - 48\,x^{3} - 18\,x^{4} - 16\,n - 16\,n x + 49\,n x^{2} + 82\,n x^{3} + 33\,n x^{4} + 8\,n^{2} + 14\,n^{2} x - 8\,n^{2} x^{2} - 26\,n^{2} x^{3} - 12\,n^{2} x^{4}\right)\,P_{n-1}(x) + \left(-12\,x - 45\,x^{2} - 63\,x^{3} - 39\,x^{4} - 9\,x^{5} + 16\,n x + 60\,n x^{2} + 84\,n x^{3} + 52\,n x^{4} + 12\,n x^{5}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2123
3210921
45459051191
5141966447843231961
642840399077706765218821595
713235642277062700861305827815511249085
8429150151231234474478588589089085025931136342970007

All 15 rows

Technical details
Node id
n-7fe445ad40415678 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-9c291abd4c2cf6e9

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,     10,       9;
   4:    5,     45,      90,       51;
   5:   14,    196,     644,      784,      323;
   6:   42,    840,    3990,     7770,     6765,      2188;
   7:  132,   3564,   22770,    62700,    86130,     58278,     15511;
   8:  429,  15015,  123123,   447447,   858858,    908908,    502593,   113634;
   9: 1430,  62920,  640640,  2938936,  7367360,  10810800,   9292920,  4343160,   853467;
  10: 4862, 262548, 3238092, 18174156, 56972916, 108432324, 128537136, 92964024, 37616427, 6536382;
%F Formula
The row polynomials satisfy (2*n^2-n+(4*n^2-2*n)*x+(2*n^2-n)*x^2)*P(n,x) = (8*n^2-16*n+6+(14*n^2-16*n)*x-(8*n^2-49*n+36)*x^2-(26*n^2-82*n+48)*x^3-(12*n^2-33*n+18)*x^4)*P(n-1,x) + ((16*n-12)*x+(60*n-45)*x^2+(84*n-63)*x^3+(52*n-39)*x^4+(12*n-9)*x^5)*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^2 - 16*n + 6 + (14*n^2 - 16*n)*x - (8*n^2 - 49*n + 36)*x^2 - (26*n^2 - 82*n + 48)*x^3 - (12*n^2 - 33*n + 18)*x^4)*P[n - 1, x] + ((16*n - 12)*x + (60*n - 45)*x^2 + (84*n - 63)*x^3 + (52*n - 39)*x^4 + (12*n - 9)*x^5)*D[P[n - 1, x], x])/(2*n^2 - n + (4*n^2 - 2*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^2 - 16*n + 6 + (14*n^2 - 16*n)*x - (8*n^2 - 49*n + 36)*x^2 - (26*n^2 - 82*n + 48)*x^3 - (12*n^2 - 33*n + 18)*x^4)*v[i - 1] + ((16*n - 12)*x + (60*n - 45)*x^2 + (84*n - 63)*x^3 + (52*n - 39)*x^4 + (12*n - 9)*x^5)*deriv(v[i - 1], x))/(2*n^2 - n + (4*n^2 - 2*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.