Real-rooted atlas

A100861 → basis A013609 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(2 + 4\,x + 2\,n x + n x^{2} + n^{2} x^{2}\right)\,P_{n}(x) = \left(-6\,x - 16\,x^{2} - 4\,x^{3} + 2\,n + 9\,n x + 8\,n x^{2} - 2\,n x^{3} + 3\,n^{2} x + 9\,n^{2} x^{2} + 4\,n^{2} x^{3} + n^{3} x^{2} + 2\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-1 - 8\,x - 22\,x^{2} - 24\,x^{3} - 8\,x^{4} - 2\,n x - 11\,n x^{2} - 20\,n x^{3} - 12\,n x^{4} - n^{2} x^{2} - 4\,n^{2} x^{3} - 4\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
011
111
2246
341216
410483694
526160180366
67660010804802236
723221845040336010816

All 15 rows

Technical details
Node id
n-91df5ecbb409319d · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-04fe6071d487f55e

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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,     4;
  3:    4,    12;
  4:   10,    48,     36;
  5:   26,   160,    180;
  6:   76,   600,   1080,    480;
  7:  232,  2184,   5040,   3360;
  8:  764,  8512,  25200,  26880,  8400;
  9: 2620, 33408, 117936, 161280, 75600;
%F Formula
The row polynomials satisfy (2+(2*n+4)*x+(n^2+n)*x^2)*P(n,x) = (2*n+(3*n^2+9*n-6)*x+(n^3+9*n^2+8*n-16)*x^2+(2*n^3+4*n^2-2*n-4)*x^3)*P(n-1,x) - (1+(2*n+8)*x+(n^2+11*n+22)*x^2+(4*n^2+20*n+24)*x^3+(4*n^2+12*n+8)*x^4)*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[((2*n + (3*n^2 + 9*n - 6)*x + (n^3 + 9*n^2 + 8*n - 16)*x^2 + (2*n^3 + 4*n^2 - 2*n - 4)*x^3)*P[n - 1, x] - (1 + (2*n + 8)*x + (n^2 + 11*n + 22)*x^2 + (4*n^2 + 20*n + 24)*x^3 + (4*n^2 + 12*n + 8)*x^4)*D[P[n - 1, x], x])/(2 + (2*n + 4)*x + (n^2 + n)*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] = ((2*n + (3*n^2 + 9*n - 6)*x + (n^3 + 9*n^2 + 8*n - 16)*x^2 + (2*n^3 + 4*n^2 - 2*n - 4)*x^3)*v[i - 1] - (1 + (2*n + 8)*x + (n^2 + 11*n + 22)*x^2 + (4*n^2 + 20*n + 24)*x^3 + (4*n^2 + 12*n + 8)*x^4)*deriv(v[i - 1], x))/(2 + (2*n + 4)*x + (n^2 + n)*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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.