Real-rooted atlas

A085881 → basis A013611 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + 2\,n x\right)\,P_{n}(x) = \left(-2 - 8\,x - 8\,x^{2} + 4\,n + 12\,n x + 8\,n x^{2} + 8\,n^{2} x + 16\,n^{2} x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
12810
21296144252
31201440432038409720
4168026880120960215040134400498960
5302406048003628800967680012096000580608031842720
6665280159667201197504004257792007983360007664025602980454402424945600
71729728048432384043589145601937295360048432384000697426329605424427008017712414720214365191040

All 15 rows

Technical details
Node id
n-fe5279eda1ee04b1 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-7cabb77b45806852

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 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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:           1;
  1:           2,            8;
  2:          12,           96,           144;
  3:         120,         1440,          4320,           3840;
  4:        1680,        26880,        120960,         215040,          134400;
  5:       30240,       604800,       3628800,        9676800,        12096000,         5806080;
  6:      665280,     15966720,     119750400,      425779200,       798336000,       766402560,       298045440;
  7:    17297280,    484323840,    4358914560,    19372953600,     48432384000,     69742632960,     54244270080,     17712414720;
  8:   518918400,  16605388800,  174356582400,   929901772800,   2905943040000,   5579410636800,   6509312409600,   4250979532800,   1195587993600;
  9: 17643225600, 635156121600, 7621873459200, 47424990412800, 177843714048000, 426824913715200, 663949865779200, 650399868518400, 365849926041600, 90333315072000;
%F Formula
The row polynomials satisfy (1+2*n*x)*P(n,x) = (4*n-2+(8*n^2+12*n-8)*x+(16*n^2+8*n-8)*x^2)*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[((4*n - 2 + (8*n^2 + 12*n - 8)*x + (16*n^2 + 8*n - 8)*x^2)*P[n - 1, x])/(1 + 2*n*x)]];
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] = ((4*n - 2 + (8*n^2 + 12*n - 8)*x + (16*n^2 + 8*n - 8)*x^2)*v[i - 1])/(1 + 2*n*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

Same fingerprint ?

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.