Real-rooted atlas

A290319 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1257
2124845105
312063610745852415
41680109442584826448994574865
53024023227269508810188727344182088452919735
66652805865984210987363979238441628996229550405221125137227545
7172972801718087047186806721646465328223633303218040719248014454101514126257547514975

All 15 rows

Technical details
Node id
n-df3cff32ba440a75 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-0873d604a54e7ea7

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,            5;
  2:          12,           48,            45;
  3:         120,          636,          1074,           585;
  4:        1680,        10944,         25848,         26448,          9945;
  5:       30240,       232272,        695088,       1018872,        734418,        208845;
  6:      665280,      5865984,      21098736,      39792384,      41628996,      22955040,       5221125;
  7:    17297280,    171808704,     718680672,    1646465328,    2236333032,    1804071924,     801445410,     151412625;
  8:   518918400,   5725071360,   27230107392,   73110422016,  121423346784,  127921147776,   83577735792,   30990077280,    4996616625;
  9: 17643225600, 213852407040, 1137651291648, 3493268322048, 6833479405248, 8841982855392, 7574725484640, 4146038851824, 1316517824610, 184874815125;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (4*n-2+(8*n-1)*x+(4*n+1)*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 - 1)*x + (4*n + 1)*x^2)*P[n - 1, x])/(1 + 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 - 1)*x + (4*n + 1)*x^2)*v[i - 1])/(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.