Real-rooted atlas

A062196 → D → D+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
133
2201232
37512030225
4204660420601344
54552520315011201057350
688875601596010920252016838016
71575191526174070560308705040252189189

All 15 rows

Technical details
Node id
n-9a888e82b6e41ae9 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-20d00a65c074974f

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:   20,    12;
  3:   75,   120,     30;
  4:  204,   660,    420,      60;
  5:  455,  2520,   3150,    1120,     105;
  6:  888,  7560,  15960,   10920,    2520,    168;
  7: 1575, 19152,  61740,   70560,   30870,   5040,    252;
  8: 2600, 42840, 196560,  341040,  246960,  75600,   9240,   360;
  9: 4059, 87120, 540540, 1330560, 1455300, 731808, 166320, 15840, 495;
%F Formula
The row polynomials satisfy (n^4-2*n^3+n^2+2*n-2+(2*n^3-4*n^2-2*n+4)*x+(n^2-1)*x^2)*P(n,x) = (n^4+2*n^3+n^2+2*n+(3*n^4-2*n^3-4*n)*x+(6*n^3-8*n^2+2*n)*x^2+3*n^2*x^3)*P(n-1,x) + ((2*n^3+1)*x-(2*n^3-4*n^2+2*n+3)*x^2-(4*n^2-4*n-3)*x^3-(2*n+1)*x^4)*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 + (3*n^4 - 2*n^3 - 4*n)*x + (6*n^3 - 8*n^2 + 2*n)*x^2 + 3*n^2*x^3)*P[n - 1, x] + ((2*n^3 + 1)*x - (2*n^3 - 4*n^2 + 2*n + 3)*x^2 - (4*n^2 - 4*n - 3)*x^3 - (2*n + 1)*x^4)*D[P[n - 1, x], x])/(n^4 - 2*n^3 + n^2 + 2*n - 2 + (2*n^3 - 4*n^2 - 2*n + 4)*x + (n^2 - 1)*x^2)]];
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 + (3*n^4 - 2*n^3 - 4*n)*x + (6*n^3 - 8*n^2 + 2*n)*x^2 + 3*n^2*x^3)*v[i - 1] + ((2*n^3 + 1)*x - (2*n^3 - 4*n^2 + 2*n + 3)*x^2 - (4*n^2 - 4*n - 3)*x^3 - (2*n + 1)*x^4)*deriv(v[i - 1], x))/(n^4 - 2*n^3 + n^2 + 2*n - 2 + (2*n^3 - 4*n^2 - 2*n + 4)*x + (n^2 - 1)*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.