Real-rooted atlas

A103633 → basis A007318 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(8 + 37\,x + 62\,x^{2} + 45\,x^{3} + 12\,x^{4} - 7\,n - 33\,n x - 57\,n x^{2} - 43\,n x^{3} - 12\,n x^{4}\right)\,P_{n-1}(x) + \left(4 + 26\,x + 66\,x^{2} + 82\,x^{3} + 50\,x^{4} + 12\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1123
227615
3211201245
4428738436225
543612923020472675
688436684710987562163375
78100534157927922952172843210125
816224136847601032114280123126048129650625
9162561816749619841349224087230672133922592151875
103256044002044062170129367186510183960118800453607776759375
113262455202924010305025370744524455698048672028296098496155522278125
126413441291275040293820816564165160924496922644380202608010458723265924665611390625
136414721560010086444390014042043284737575291075437647314840509803224183366998409331234171875
141283136356162485281190392414010810782086213600313234625837260972321405842013076886546882286144279936170859375
Technical details
Node id
n-b5d38d742ea2388f · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-2e6f63c0c32e8505

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:  1,   2;
  2:  2,   7,    6;
  3:  2,  11,   20,   12;
  4:  4,  28,   73,   84,    36;
  5:  4,  36,  129,  230,   204,    72;
  6:  8,  84,  366,  847,  1098,   756,   216;
  7:  8, 100,  534, 1579,  2792,  2952,  1728,   432;
  8: 16, 224, 1368, 4760, 10321, 14280, 12312,  6048,  1296;
  9: 16, 256, 1816, 7496, 19841, 34922, 40872, 30672, 13392, 2592;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = -(7*n-8+(33*n-37)*x+(57*n-62)*x^2+(43*n-45)*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) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(7*n - 8 + (33*n - 37)*x + (57*n - 62)*x^2 + (43*n - 45)*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(7*n - 8 + (33*n - 37)*x + (57*n - 62)*x^2 + (43*n - 45)*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

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.