Real-rooted atlas

A166962 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(1 + 8\,x + 23\,x^{2} + 32\,x^{3} + 22\,x^{4} + 6\,x^{5} - 4\,n x - 21\,n x^{2} - 39\,n x^{3} - 31\,n x^{4} - 9\,n x^{5} + 3\,n^{2} x^{2} + 9\,n^{2} x^{3} + 9\,n^{2} x^{4} + 3\,n^{2} x^{5}\right)\,P_{n-1}(x) + \left(7\,x + 37\,x^{2} + 81\,x^{3} + 91\,x^{4} + 52\,x^{5} + 12\,x^{6} - 6\,n x^{2} - 24\,n x^{3} - 36\,n x^{4} - 24\,n x^{5} - 6\,n x^{6}\right)\,P_{n-1}'(x) + \left(3\,x^{2} + 15\,x^{3} + 30\,x^{4} + 30\,x^{5} + 15\,x^{6} + 3\,x^{7}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1123
21141429
31106278174559
4183846787162332216001
5166908066022968024546489756652251
6153502148651070633401308716010668248321142035570181
71427994290231942206352906356697408694974485720006041461328082473387079

All 15 rows

Technical details
Node id
n-d33cbe5548dd9a88 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-2958c20fa270b165

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,       14,          14;
  3: 1,      106,         278,          174;
  4: 1,      838,        4678,         7162,          3322;
  5: 1,     6690,       80660,       229680,        245464,         89756;
  6: 1,    53502,     1486510,      7063340,      13087160,      10668248,       3211420;
  7: 1,   427994,    29023194,    220635290,     635669740,     869497448,     572000604,     146132808;
  8: 1,  3423926,   587659478,   7154906294,   30406730050,   62298056248,   67280173924,   37020644528,    8202467544;
  9: 1, 27391378, 12131989688, 242901190412, 1479654376772, 4277047855828, 6755155744492, 6010368385792, 2837891772656, 554489060200;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (1-(4*n-8)*x+(3*n^2-21*n+23)*x^2+(9*n^2-39*n+32)*x^3+(9*n^2-31*n+22)*x^4+(3*n^2-9*n+6)*x^5)*P(n-1,x) + (7*x-(6*n-37)*x^2-(24*n-81)*x^3-(36*n-91)*x^4-(24*n-52)*x^5-(6*n-12)*x^6)*d/dx P(n-1,x) + (3*x^2+15*x^3+30*x^4+30*x^5+15*x^6+3*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 - (4*n - 8)*x + (3*n^2 - 21*n + 23)*x^2 + (9*n^2 - 39*n + 32)*x^3 + (9*n^2 - 31*n + 22)*x^4 + (3*n^2 - 9*n + 6)*x^5)*P[n - 1, x] + (7*x - (6*n - 37)*x^2 - (24*n - 81)*x^3 - (36*n - 91)*x^4 - (24*n - 52)*x^5 - (6*n - 12)*x^6)*D[P[n - 1, x], x] + (3*x^2 + 15*x^3 + 30*x^4 + 30*x^5 + 15*x^6 + 3*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 - (4*n - 8)*x + (3*n^2 - 21*n + 23)*x^2 + (9*n^2 - 39*n + 32)*x^3 + (9*n^2 - 31*n + 22)*x^4 + (3*n^2 - 9*n + 6)*x^5)*v[i - 1] + (7*x - (6*n - 37)*x^2 - (24*n - 81)*x^3 - (36*n - 91)*x^4 - (24*n - 52)*x^5 - (6*n - 12)*x^6)*deriv(v[i - 1], x) + (3*x^2 + 15*x^3 + 30*x^4 + 30*x^5 + 15*x^6 + 3*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.