Real-rooted atlas

A035469 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(-2 - x + 3\,x^{2} + 2\,x^{3} + 3\,n + 5\,n x + n x^{2} - n x^{3}\right)\,P_{n-1}(x) + \left(x + 3\,x^{2} + 3\,x^{3} + x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2459
3286841137
4280100011844652929
53640170802992023200672180561
6582403376807802008982005153801179412707641
711065607627200218335603323348028375340128879242433145107497209
82434432019423712066228288012512057601414794640957657008359351328576732814921546337

All 15 rows

Technical details
Node id
n-0da1732cb46801e3 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-7b46e765e8c58e5d

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:           4,            5;
   3:          28,           68,           41;
   4:         280,         1000,         1184,           465;
   5:        3640,        17080,        29920,         23200,          6721;
   6:       58240,       337680,       780200,        898200,        515380,        117941;
   7:     1106560,      7627200,     21833560,      33233480,      28375340,      12887924,       2433145;
   8:    24344320,    194237120,    662282880,    1251205760,    1414794640,     957657008,     359351328,      57673281;
   9:   608608000,   5512707200,  21790207040,   49100844800,   68997774160,   61924342480,   34667560480,   11070122144,   1543866945;
  10: 17041024000, 172649276800, 775629612800, 2028292414400, 3402910340480, 3798934528480, 2822360576560, 1345710743440, 373698364100, 46052954821;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (3*n-2+(5*n-1)*x+(n+3)*x^2-(n-2)*x^3)*P(n-1,x) + (x+3*x^2+3*x^3+x^4)*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[((3*n - 2 + (5*n - 1)*x + (n + 3)*x^2 - (n - 2)*x^3)*P[n - 1, x] + (x + 3*x^2 + 3*x^3 + x^4)*D[P[n - 1, x], x])/(1 + 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] = ((3*n - 2 + (5*n - 1)*x + (n + 3)*x^2 - (n - 2)*x^3)*v[i - 1] + (x + 3*x^2 + 3*x^3 + x^4)*deriv(v[i - 1], x))/(1 + 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

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.