Real-rooted atlas

A145596 → x ↦ x+1 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(4\,n + 8\,n x + 4\,n x^{2} + n^{2} + 2\,n^{2} x + n^{2} x^{2}\right)\,P_{n}(x) = \left(6 + 18\,x + 14\,x^{2} - 2\,x^{3} - 4\,x^{4} + 10\,n + 35\,n x + 40\,n x^{2} + 15\,n x^{3} + 4\,n^{2} + 17\,n^{2} x + 26\,n^{2} x^{2} + 17\,n^{2} x^{3} + 4\,n^{2} x^{4}\right)\,P_{n-1}(x) + \left(-2\,x - 10\,x^{2} - 18\,x^{3} - 14\,x^{4} - 4\,x^{5} - 2\,n x - 10\,n x^{2} - 18\,n x^{3} - 14\,n x^{4} - 4\,n x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
14610
214423187
348216320156740
4165990220521607856305
55724290127601881013740397854150
6200218018670671321321452998454620335469399
7707274256331968819000120411210548725098241048564105960

All 15 rows

Technical details
Node id
n-775257ac5e538130 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-be1a721ff0cae911

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:     1;
  1:     4,       6;
  2:    14,      42,      31;
  3:    48,     216,     320,      156;
  4:   165,     990,    2205,     2160,      785;
  5:   572,    4290,   12760,    18810,    13740,     3978;
  6:  2002,   18018,   67067,   132132,   145299,    84546,    20335;
  7:  7072,   74256,  331968,   819000,  1204112,  1054872,   509824,   104856;
  8: 25194,  302328, 1577940,  4678128,  8615880, 10093356,  7344246,  3034512,   545073;
  9: 90440, 1220940, 7286880, 25232760, 55863360, 81995760, 79786560, 49627620, 17904180, 2854350;
%F Formula
The row polynomials satisfy (n^2+4*n+(2*n^2+8*n)*x+(n^2+4*n)*x^2)*P(n,x) = (4*n^2+10*n+6+(17*n^2+35*n+18)*x+(26*n^2+40*n+14)*x^2+(17*n^2+15*n-2)*x^3+(4*n^2-4)*x^4)*P(n-1,x) - ((2*n+2)*x+(10*n+10)*x^2+(18*n+18)*x^3+(14*n+14)*x^4+(4*n+4)*x^5)*d/dx 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 + 10*n + 6 + (17*n^2 + 35*n + 18)*x + (26*n^2 + 40*n + 14)*x^2 + (17*n^2 + 15*n - 2)*x^3 + (4*n^2 - 4)*x^4)*P[n - 1, x] - ((2*n + 2)*x + (10*n + 10)*x^2 + (18*n + 18)*x^3 + (14*n + 14)*x^4 + (4*n + 4)*x^5)*D[P[n - 1, x], x])/(n^2 + 4*n + (2*n^2 + 8*n)*x + (n^2 + 4*n)*x^2)]];
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 + 10*n + 6 + (17*n^2 + 35*n + 18)*x + (26*n^2 + 40*n + 14)*x^2 + (17*n^2 + 15*n - 2)*x^3 + (4*n^2 - 4)*x^4)*v[i - 1] - ((2*n + 2)*x + (10*n + 10)*x^2 + (18*n + 18)*x^3 + (14*n + 14)*x^4 + (4*n + 4)*x^5)*deriv(v[i - 1], x))/(n^2 + 4*n + (2*n^2 + 8*n)*x + (n^2 + 4*n)*x^2)); 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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.