Real-rooted atlas

A166960 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(1 + 6\,x + 13\,x^{2} + 14\,x^{3} + 8\,x^{4} + 2\,x^{5} - 2\,n x - 9\,n x^{2} - 15\,n x^{3} - 11\,n x^{4} - 3\,n x^{5} + n^{2} x^{2} + 3\,n^{2} x^{3} + 3\,n^{2} x^{4} + n^{2} x^{5}\right)\,P_{n-1}(x) + \left(3\,x + 15\,x^{2} + 31\,x^{3} + 33\,x^{4} + 18\,x^{5} + 4\,x^{6} - 2\,n x^{2} - 8\,n x^{3} - 12\,n x^{4} - 8\,n x^{5} - 2\,n x^{6}\right)\,P_{n-1}'(x) + \left(x^{2} + 5\,x^{3} + 10\,x^{4} + 10\,x^{5} + 5\,x^{6} + x^{7}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1123
218817
31307850159
411166129384422109
5145847001310014068521237539
6118243727217018831389625752078364859065
7172863081062201230624530085379765648836145575224404487

All 15 rows

Technical details
Node id
n-b029531a92200281 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-664a50ba99800b0d

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: 1,      2;
  2: 1,      8,        8;
  3: 1,     30,       78,        50;
  4: 1,    116,      612,       938,        442;
  5: 1,    458,     4700,     13100,      14068,       5212;
  6: 1,   1824,    37272,    170188,     313896,     257520,       78364;
  7: 1,   7286,   308106,   2201230,    6245300,    8537976,     5648836,    1455752;
  8: 1,  29132,  2634548,  29101318,  120251570,  244523456,   264314212,  146162552,   32616128;
  9: 1, 116514, 23040936, 397530744, 2322375192, 6616332516, 10410902244, 9276631152, 4398842640, 864550088;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (1-(2*n-6)*x+(n^2-9*n+13)*x^2+(3*n^2-15*n+14)*x^3+(3*n^2-11*n+8)*x^4+(n^2-3*n+2)*x^5)*P(n-1,x) + (3*x-(2*n-15)*x^2-(8*n-31)*x^3-(12*n-33)*x^4-(8*n-18)*x^5-(2*n-4)*x^6)*d/dx P(n-1,x) + (x^2+5*x^3+10*x^4+10*x^5+5*x^6+x^7)*d^2/dx^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[((1 - (2*n - 6)*x + (n^2 - 9*n + 13)*x^2 + (3*n^2 - 15*n + 14)*x^3 + (3*n^2 - 11*n + 8)*x^4 + (n^2 - 3*n + 2)*x^5)*P[n - 1, x] + (3*x - (2*n - 15)*x^2 - (8*n - 31)*x^3 - (12*n - 33)*x^4 - (8*n - 18)*x^5 - (2*n - 4)*x^6)*D[P[n - 1, x], x] + (x^2 + 5*x^3 + 10*x^4 + 10*x^5 + 5*x^6 + x^7)*D[P[n - 1, x], {x, 2}])/(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] = ((1 - (2*n - 6)*x + (n^2 - 9*n + 13)*x^2 + (3*n^2 - 15*n + 14)*x^3 + (3*n^2 - 11*n + 8)*x^4 + (n^2 - 3*n + 2)*x^5)*v[i - 1] + (3*x - (2*n - 15)*x^2 - (8*n - 31)*x^3 - (12*n - 33)*x^4 - (8*n - 18)*x^5 - (2*n - 4)*x^6)*deriv(v[i - 1], x) + (x^2 + 5*x^3 + 10*x^4 + 10*x^5 + 5*x^6 + x^7)*deriv(deriv(v[i - 1], x), 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

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.