Real-rooted atlas

A048854 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-6 - x + n x\right)\,P_{n}(x) = \left(36 - 6\,x - 24\,n - 10\,n x + n x^{2} + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(-24\,x + 4\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2224
31224339
4120360904574
5168067202520224511149
63024015120075600100804506267576
7665280399168024948004435202970079277625779
817297280121080960908107202018016018018007207212748251244274

All 15 rows

Technical details
Node id
n-be026e31b07cd533 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-b8ec366cca2adfe1

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,            2;
   3:          12,           24,            3;
   4:         120,          360,           90,           4;
   5:        1680,         6720,         2520,         224,          5;
   6:       30240,       151200,        75600,       10080,        450,         6;
   7:      665280,      3991680,      2494800,      443520,      29700,       792,        7;
   8:    17297280,    121080960,     90810720,    20180160,    1801800,     72072,     1274,      8;
   9:   518918400,   4151347200,   3632428800,   968647680,  108108000,   5765760,   152880,   1920,    9;
  10: 17643225600, 158789030400, 158789030400, 49401031680, 6616209600, 441080640, 15593760, 293760, 2754, 10;
%F Formula
The row polynomials satisfy -(6-(n-1)*x)*P(n,x) = -(24*n-36-(4*n^2-10*n-6)*x-n*x^2)*P(n-1,x) - (24*x-4*n*x^2)*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[(-(24*n - 36 - (4*n^2 - 10*n - 6)*x - n*x^2)*P[n - 1, x] - (24*x - 4*n*x^2)*D[P[n - 1, x], x])/(-6 + (n - 1)*x)]];
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] = (-(24*n - 36 - (4*n^2 - 10*n - 6)*x - n*x^2)*v[i - 1] - (24*x - 4*n*x^2)*deriv(v[i - 1], x))/(-6 + (n - 1)*x)); 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.