Real-rooted atlas

A075499 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-x^{2}\right)\,P_{n-2}(x) + \left(-4\,x^{2}\right)\,P_{n-2}'(x) + \left(4 + 2\,x\right)\,P_{n-1}(x) + \left(4\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 16 + 24\,x + 3\,x^{2}\)
\(P_{1}(x) = 64 + 224\,x + 72\,x^{2} + 4\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
01624343
164224724364
22561920120016053541
3102415872172804160300638642
4409612902423116889600112005047465599
5163841040384296755217418243360002553678486128472
6655368355840371712003182592088972801016064517441152987384745
726214466977792458588160558776320217728000350622722634240960001620101340126558

All 15 rows

Technical details
Node id
n-e45ad4ad2f3385b7 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-c2c2fa53afb63ab8

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:      16,         24,           3;
  1:      64,        224,          72,            4;
  2:     256,       1920,        1200,          160,            5;
  3:    1024,      15872,       17280,         4160,          300,           6;
  4:    4096,     129024,      231168,        89600,        11200,         504,          7;
  5:   16384,    1040384,     2967552,      1741824,       336000,       25536,        784,         8;
  6:   65536,    8355840,    37171200,     31825920,      8897280,     1016064,      51744,      1152,        9;
  7:  262144,   66977792,   458588160,    558776320,    217728000,    35062272,    2634240,     96000,     1620,     10;
  8: 1048576,  536346624,  5603524608,   9551872000,   5053030400,  1102768128,  114664704,   6082560,   166320,   2200,   11;
  9: 4194304, 4292870144, 68046815232, 160301318144, 113000448000, 32530071552, 4497174528, 325687296, 12830400, 272800, 2904, 12;
%F Formula
The row polynomials satisfy P(n,x) = -x^2*P(n-2,x) - 4*x^2*d/dx P(n-2,x) + (4+2*x)*P(n-1,x) + 4*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 16 + 24*x + 3*x^2, P(1,x) = 64 + 224*x + 72*x^2 + 4*x^3.
%t Mathematica
P[0, x_] := 16 + 24*x + 3*x^2;
P[1, x_] := 64 + 224*x + 72*x^2 + 4*x^3;
P[n_, x_] := P[n, x] = Expand[-x^2*P[n - 2, x] - 4*x^2*D[P[n - 2, x], x] + (4 + 2*x)*P[n - 1, x] + 4*x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 16 + 24*x + 3*x^2; v[2] = 64 + 224*x + 72*x^2 + 4*x^3; for(n = 2, N, my(i = n + 1); v[i] = -x^2*v[i - 2] - 4*x^2*deriv(v[i - 2], x) + (4 + 2*x)*v[i - 1] + 4*x*deriv(v[i - 1], x)); 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. [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. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.