Real-rooted atlas

A064861 → reverse → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(-5 - 26\,x - 49\,x^{2} - 40\,x^{3} - 12\,x^{4} - 7\,n - 33\,n x - 57\,n x^{2} - 43\,n x^{3} - 12\,n x^{4}\right)\,P_{n-1}(x) + \left(4 + 26\,x + 66\,x^{2} + 82\,x^{3} + 50\,x^{4} + 12\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 2 + 7\,x + 6\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
027615
1420331875
2428738436225
38682303873241081125
488436684710987562163375
516192984279247374806270064816875
616224136847601032114280123126048129650625
73249634081362434922595236746449032207363888253125

All 15 rows

Technical details
Node id
n-22aa0691270f8663 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-a7854389020f6acb

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:  2,    7,     6;
  1:  4,   20,    33,    18;
  2:  4,   28,    73,    84,     36;
  3:  8,   68,   230,   387,    324,    108;
  4:  8,   84,   366,   847,   1098,    756,    216;
  5: 16,  192,   984,  2792,   4737,   4806,   2700,    648;
  6: 16,  224,  1368,  4760,  10321,  14280,  12312,   6048,   1296;
  7: 32,  496,  3408, 13624,  34922,  59523,  67464,  49032,  20736,   3888;
  8: 32,  560,  4400, 20440,  62170, 129367, 186510, 183960, 118800,  45360,   7776;
  9: 64, 1216, 10480, 54080, 185660, 445244, 761121, 927450, 789480, 447120, 151632, 23328;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = -(7*n+5+(33*n+26)*x+(57*n+49)*x^2+(43*n+40)*x^3+(12*n+12)*x^4)*P(n-1,x) + (4+26*x+66*x^2+82*x^3+50*x^4+12*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 2 + 7*x + 6*x^2.
%t Mathematica
P[0, x_] := 2 + 7*x + 6*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(7*n + 5 + (33*n + 26)*x + (57*n + 49)*x^2 + (43*n + 40)*x^3 + (12*n + 12)*x^4)*P[n - 1, x] + (4 + 26*x + 66*x^2 + 82*x^3 + 50*x^4 + 12*x^5)*D[P[n - 1, x], x])/(1 + 2*x + x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 2 + 7*x + 6*x^2; for(n = 1, N, my(i = n + 1); v[i] = (-(7*n + 5 + (33*n + 26)*x + (57*n + 49)*x^2 + (43*n + 40)*x^3 + (12*n + 12)*x^4)*v[i - 1] + (4 + 26*x + 66*x^2 + 82*x^3 + 50*x^4 + 12*x^5)*deriv(v[i - 1], x))/(1 + 2*x + 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

Candidate equalities ?

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.