Real-rooted atlas

A243662 → D

nonnegative coefficients

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

Recurrence

\(\left(-4\,n - 6\,n^{2} + 4\,n^{3}\right)\,P_{n}(x) = \left(-6 + 33\,n + 16\,n x - 54\,n^{2} - 32\,n^{2} x + 27\,n^{3} + 12\,n^{3} x\right)\,P_{n-1}(x) + \left(-2\,x + 9\,n x + 6\,n x^{2} - 9\,n^{2} x - 8\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
10
211
38210
45530388
5364312724752
623802800105014056375
71550423256122402720240653966
81009471843381256853990059853787457240

All 15 rows

Technical details
Node id
n-60165740581e38e9 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-ba20aaaea70961b4

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 -2)
  • 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:       0;
   2:       1;
   3:       8,        2;
   4:      55,       30,        3;
   5:     364,      312,       72,       4;
   6:    2380,     2800,     1050,     140,       5;
   7:   15504,    23256,    12240,    2720,     240,      6;
   8:  100947,   184338,   125685,   39900,    5985,    378,     7;
   9:  657800,  1416800,  1190112,  495880,  107800,  11760,   560,   8;
  10: 4292145, 10656360, 10656360, 5525520, 1593900, 255024, 21252, 792, 9;
%F Formula
The row polynomials satisfy (4*n^3-6*n^2-4*n)*P(n,x) = (27*n^3-54*n^2+33*n-6+(12*n^3-32*n^2+16*n)*x)*P(n-1,x) - ((9*n^2-9*n+2)*x+(8*n^2-6*n)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((27*n^3 - 54*n^2 + 33*n - 6 + (12*n^3 - 32*n^2 + 16*n)*x)*P[n - 1, x] - ((9*n^2 - 9*n + 2)*x + (8*n^2 - 6*n)*x^2)*D[P[n - 1, x], x])/(4*n^3 - 6*n^2 - 4*n)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = ((27*n^3 - 54*n^2 + 33*n - 6 + (12*n^3 - 32*n^2 + 16*n)*x)*v[i - 1] - ((9*n^2 - 9*n + 2)*x + (8*n^2 - 6*n)*x^2)*deriv(v[i - 1], x))/(4*n^3 - 6*n^2 - 4*n)); 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. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.