Real-rooted atlas

A269945 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(3\,x + 14\,x^{2} + 25\,x^{3} + 20\,x^{4} + 6\,x^{5} - n x - 8\,n x^{2} - 18\,n x^{3} - 16\,n x^{4} - 5\,n x^{5} + n^{2} x^{2} + 3\,n^{2} x^{3} + 3\,n^{2} x^{4} + n^{2} x^{5}\right)\,P_{n-1}(x) + \left(x + 10\,x^{2} + 30\,x^{3} + 40\,x^{4} + 25\,x^{5} + 6\,x^{6} - 2\,n x^{2} - 8\,n x^{3} - 12\,n x^{4} - 8\,n x^{5} - 2\,n x^{6}\right)\,P_{n-1}'(x) + \left(x^{2} + 5\,x^{3} + 10\,x^{4} + 10\,x^{5} + 5\,x^{6} + x^{7}\right)\,P_{n-1}''(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2011
30123
4017715
501245937121
601894085832641345
701346278269076902243319371
801137119853771621280659729827913351663

All 15 rows

Technical details
Node id
n-ccc32210e94923b3 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-4c47ff87ed3f3de6

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1;
   2: 0, 1;
   3: 0, 1,     2;
   4: 0, 1,     7,       7;
   5: 0, 1,    24,      59,       37;
   6: 0, 1,    89,     408,      583,      264;
   7: 0, 1,   346,    2782,     6907,     6902,     2433;
   8: 0, 1,  1371,   19853,    77162,   128065,    97298,    27913;
   9: 0, 1,  5468,  151269,   870413,  2139777,  2632822,  1604561,   386906;
  10: 0, 1, 21853, 1224742, 10144925, 34851583, 61938621, 60255508, 30572553, 6346119;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = -((n-3)*x-(n^2-8*n+14)*x^2-(3*n^2-18*n+25)*x^3-(3*n^2-16*n+20)*x^4-(n^2-5*n+6)*x^5)*P(n-1,x) + (x-(2*n-10)*x^2-(8*n-30)*x^3-(12*n-40)*x^4-(8*n-25)*x^5-(2*n-6)*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 >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-((n - 3)*x - (n^2 - 8*n + 14)*x^2 - (3*n^2 - 18*n + 25)*x^3 - (3*n^2 - 16*n + 20)*x^4 - (n^2 - 5*n + 6)*x^5)*P[n - 1, x] + (x - (2*n - 10)*x^2 - (8*n - 30)*x^3 - (12*n - 40)*x^4 - (8*n - 25)*x^5 - (2*n - 6)*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, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-((n - 3)*x - (n^2 - 8*n + 14)*x^2 - (3*n^2 - 18*n + 25)*x^3 - (3*n^2 - 16*n + 20)*x^4 - (n^2 - 5*n + 6)*x^5)*v[i - 1] + (x - (2*n - 10)*x^2 - (8*n - 30)*x^3 - (12*n - 40)*x^4 - (8*n - 25)*x^5 - (2*n - 6)*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, v[i])))};
T(10)

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.