Real-rooted atlas

A269947 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(-1 - x + 3\,n + 6\,n x + 3\,n x^{2} - 3\,n^{2} - 6\,n^{2} x - 3\,n^{2} x^{2} + n^{3} + 2\,n^{3} x + n^{3} x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1011
20123
308251851
4021689911865042805
50138247157613433910934632760361845
60172800010688824258109513059496417872596412776090823095
70373248000268376198478946402241220948859110499587924476994949289572392039326400135

All 15 rows

Technical details
Node id
n-a97ecc679ffc37ba · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-411590dcfce40f62

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: 0,              1;
  2: 0,              1,               2;
  3: 0,              8,              25,               18;
  4: 0,            216,             899,             1186,              504;
  5: 0,          13824,           71576,           134339,           109346,            32760;
  6: 0,        1728000,        10688824,         25810951,         30594964,         17872596,          4127760;
  7: 0,      373248000,      2683761984,       7894640224,      12209488591,      10499587924,       4769949492,        895723920;
  8: 0,   128024064000,   1048927672512,    3631075719328,    6903610823769,    7801422733236,    5247950921612,    1948095929808,     308129028480;
  9: 0, 65548320768000, 602727313158144, 2397210664294592, 5397390585784992, 7535880792010329, 6689080734015412, 3689623938848652, 1157135274573264, 158070191610240;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (n^3-3*n^2+3*n-1+(2*n^3-6*n^2+6*n-1)*x+(n^3-3*n^2+3*n)*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[((n^3 - 3*n^2 + 3*n - 1 + (2*n^3 - 6*n^2 + 6*n - 1)*x + (n^3 - 3*n^2 + 3*n)*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] = ((n^3 - 3*n^2 + 3*n - 1 + (2*n^3 - 6*n^2 + 6*n - 1)*x + (n^3 - 3*n^2 + 3*n)*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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.