Real-rooted atlas

A085881 → basis A013611 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
16410
26014448252
3840360043209609720
4151209408020160016128026880498960
53326402721600846720012096000725760096768031842720
6864864087816960359251200745113600798336000383201280425779202424945600
7259459200314810496015982686720435891456006780533760058118860800232475443202214051840214365191040

All 15 rows

Technical details
Node id
n-bf0bee765dd426d0 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-4be2d3b20c0d29e0

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:            6,             4;
  2:           60,           144,             48;
  3:          840,          3600,           4320,             960;
  4:        15120,         94080,         201600,          161280,           26880;
  5:       332640,       2721600,        8467200,        12096000,         7257600,          967680;
  6:      8648640,      87816960,      359251200,       745113600,       798336000,       383201280,        42577920;
  7:    259459200,    3148104960,    15982686720,     43589145600,     67805337600,     58118860800,     23247544320,      2214051840;
  8:   8821612800,  124540416000,   755545190400,   2557229875200,   5230697472000,   6509312409600,   4649508864000,   1594117324800,    132843110400;
  9: 335221286400, 5398827033600, 38109367296000, 154131218841600, 391256170905600, 640237370572800, 663949865779200, 406499917824000, 121949975347200, 9033331507200;
%F Formula
The row polynomials satisfy (2*n-1+2*x)*P(n,x) = (8*n^2-2+(16*n^2+8*n-8)*x+(16*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[((8*n^2 - 2 + (16*n^2 + 8*n - 8)*x + (16*n - 8)*x^2)*P[n - 1, x])/(2*n - 1 + 2*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] = ((8*n^2 - 2 + (16*n^2 + 8*n - 8)*x + (16*n - 8)*x^2)*v[i - 1])/(2*n - 1 + 2*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.