Real-rooted atlas

A292219 → D+1

nonnegative coefficients

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

Recurrence

\(\left(4 - 3\,x + n\right)\,P_{n}(x) = \left(-10 + 7\,x - 3\,x^{2} + 18\,n - 11\,n x + 4\,n^{2}\right)\,P_{n-1}(x) + \left(4\,x - 12\,x^{2} + 4\,n x\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2718
380221103
412605044511810
52520013104172876140109
660984038808067320440011511069756
7172972801297296027799202402409360162133299923
8562161600484323840123243120132132006825001764021711183642118

All 15 rows

Technical details
Node id
n-f8039dbc976d8dfa · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-a38c1fc9c9cd2b68

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:            7,            1;
   3:           80,           22,            1;
   4:         1260,          504,           45,           1;
   5:        25200,        13104,         1728,          76,          1;
   6:       609840,       388080,        67320,        4400,        115,         1;
   7:     17297280,     12972960,      2779920,      240240,       9360,       162,       1;
   8:    562161600,    484323840,    123243120,    13213200,     682500,     17640,     217,     1;
   9:  20583763200,  19995655680,   5881075200,   751470720,   48266400,   1656480,   30464,   280,   1;
  10: 838053216000, 905097473280, 301699157760, 44696171520, 3428399520, 146512800, 3581424, 49248, 351, 1;
%F Formula
The row polynomials satisfy (n+4-3*x)*P(n,x) = (4*n^2+18*n-10-(11*n-7)*x-3*x^2)*P(n-1,x) + ((4*n+4)*x-12*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[((4*n^2 + 18*n - 10 - (11*n - 7)*x - 3*x^2)*P[n - 1, x] + ((4*n + 4)*x - 12*x^2)*D[P[n - 1, x], x])/(n + 4 - 3*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] = ((4*n^2 + 18*n - 10 - (11*n - 7)*x - 3*x^2)*v[i - 1] + ((4*n + 4)*x - 12*x^2)*deriv(v[i - 1], x))/(n + 4 - 3*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.