Real-rooted atlas

A109983 → D+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
11528
2021930657
30034916215020384
400041015601070700702505
500005181150046906230315025216008
6000006295340215540331803364213860924100849
70000007449686042756131250211134172788600603432628736

All 15 rows

Technical details
Node id
n-45f4a43cf1c73c51 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-97a44f7e6824c858

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, intercept 0)
  • 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: 1;
  1: 1, 5,  2;
  2: 0, 2, 19, 30,   6;
  3: 0, 0,  3, 49, 162, 150,   20;
  4: 0, 0,  0,  4, 101, 560, 1070,  700,    70;
  5: 0, 0,  0,  0,   5, 181, 1500, 4690,  6230,  3150,    252;
  6: 0, 0,  0,  0,   0,   6,  295, 3402, 15540, 33180,  33642,  13860,     924;
  7: 0, 0,  0,  0,   0,   0,    7,  449,  6860, 42756, 131250, 211134,  172788,   60060,    3432;
  8: 0, 0,  0,  0,   0,   0,    0,    8,   649, 12672, 102900, 425040,  971586, 1249248,  856284,  257400,   12870;
  9: 0, 0,  0,  0,   0,   0,    0,    0,     9,   901,  21870, 223740, 1189650, 3621618, 6558552, 7009860, 4131270, 1093950, 48620;
%F Formula
The row polynomials satisfy (n^3-2*n^2+n-n*x+(2*n^2-3*n)*x^2+n*x^3)*P(n,x) = -((n^3-6*n^2+6*n)*x-(6*n^2-8*n-4)*x^2+(2*n^2-2*n+6)*x^3-3*n*x^4-2*x^5)*P(n-1,x) + ((2*n^2-3*n)*x^2+(2*n^2-3*n-2)*x^3+(4*n-2)*x^4+(4*n+2)*x^5+2*x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-((n^3 - 6*n^2 + 6*n)*x - (6*n^2 - 8*n - 4)*x^2 + (2*n^2 - 2*n + 6)*x^3 - 3*n*x^4 - 2*x^5)*P[n - 1, x] + ((2*n^2 - 3*n)*x^2 + (2*n^2 - 3*n - 2)*x^3 + (4*n - 2)*x^4 + (4*n + 2)*x^5 + 2*x^6)*D[P[n - 1, x], x])/(n^3 - 2*n^2 + n - n*x + (2*n^2 - 3*n)*x^2 + n*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-((n^3 - 6*n^2 + 6*n)*x - (6*n^2 - 8*n - 4)*x^2 + (2*n^2 - 2*n + 6)*x^3 - 3*n*x^4 - 2*x^5)*v[i - 1] + ((2*n^2 - 3*n)*x^2 + (2*n^2 - 3*n - 2)*x^3 + (4*n - 2)*x^4 + (4*n + 2)*x^5 + 2*x^6)*deriv(v[i - 1], x))/(n^3 - 2*n^2 + n - n*x + (2*n^2 - 3*n)*x^2 + n*x^3)); 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

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.