Real-rooted atlas

A281260 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(2\,n + 4\,n x + 2\,n x^{2} + n^{2} + 2\,n^{2} x + n^{2} x^{2}\right)\,P_{n}(x) = \left(-1 - x + x^{2} + x^{3} + 3\,n x + 6\,n x^{2} + 3\,n x^{3} + n^{2} + 4\,n^{2} x + 5\,n^{2} x^{2} + 2\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(x + 3\,x^{2} + 3\,x^{3} + x^{4} + 2\,n x + 6\,n x^{2} + 6\,n x^{3} + 2\,n x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1022
20257
302121428
402215642121
50232144240132550
602453008259904292591
70260550220042904004143012536

All 15 rows

Technical details
Node id
n-5adb183b9026eb81 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-63d66b489de6a076

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: 0, 2;
  2: 0, 2,  5;
  3: 0, 2, 12,   14;
  4: 0, 2, 21,   56,    42;
  5: 0, 2, 32,  144,   240,   132;
  6: 0, 2, 45,  300,   825,   990,   429;
  7: 0, 2, 60,  550,  2200,  4290,  4004,  1430;
  8: 0, 2, 77,  924,  5005, 14014, 21021, 16016,  4862;
  9: 0, 2, 96, 1456, 10192, 38220, 81536, 99008, 63648, 16796;
%F Formula
The row polynomials satisfy (n^2+2*n+(2*n^2+4*n)*x+(n^2+2*n)*x^2)*P(n,x) = (n^2-1+(4*n^2+3*n-1)*x+(5*n^2+6*n+1)*x^2+(2*n^2+3*n+1)*x^3)*P(n-1,x) + ((2*n+1)*x+(6*n+3)*x^2+(6*n+3)*x^3+(2*n+1)*x^4)*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[((n^2 - 1 + (4*n^2 + 3*n - 1)*x + (5*n^2 + 6*n + 1)*x^2 + (2*n^2 + 3*n + 1)*x^3)*P[n - 1, x] + ((2*n + 1)*x + (6*n + 3)*x^2 + (6*n + 3)*x^3 + (2*n + 1)*x^4)*D[P[n - 1, x], x])/(n^2 + 2*n + (2*n^2 + 4*n)*x + (n^2 + 2*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] = ((n^2 - 1 + (4*n^2 + 3*n - 1)*x + (5*n^2 + 6*n + 1)*x^2 + (2*n^2 + 3*n + 1)*x^3)*v[i - 1] + ((2*n + 1)*x + (6*n + 3)*x^2 + (6*n + 3)*x^3 + (2*n + 1)*x^4)*deriv(v[i - 1], x))/(n^2 + 2*n + (2*n^2 + 4*n)*x + (n^2 + 2*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.