A105070 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-n + n^{2} x + n^{3} x\right)\,P_{n}(x) = \left(-1 + 4\,x - 4\,x^{2} - n - 3\,n x - 2\,n x^{2} + 2\,n^{2} x + 4\,n^{2} x^{2} + n^{3} x + 2\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-2\,x + 8\,x^{2} - 8\,x^{3} + 6\,n x^{2} - 12\,n x^{3} + 2\,n^{2} x^{2} - 4\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	P_n(1)
0	1				1
1	2				2
2	3	4			7
3	4	16			20
4	5	40	12		57
5	6	80	72		158
6	7	140	252	32	431
7	8	224	672	256	1160

All 15 rows

Technical details

Node id: n-460f9e4e6c14cb8b · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-04977a4833a7e647

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:  2;
  2:  3,   4;
  3:  4,  16;
  4:  5,  40,   12;
  5:  6,  80,   72;
  6:  7, 140,  252,   32;
  7:  8, 224,  672,  256;
  8:  9, 336, 1512, 1152,  80;
  9: 10, 480, 3024, 3840, 800;

%F Formula

The row polynomials satisfy -(n-(n^3+n^2)*x)*P(n,x) = -(n+1-(n^3+2*n^2-3*n+4)*x-(2*n^3+4*n^2-2*n-4)*x^2)*P(n-1,x) - (2*x-(2*n^2+6*n+8)*x^2+(4*n^2+12*n+8)*x^3)*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[(-(n + 1 - (n^3 + 2*n^2 - 3*n + 4)*x - (2*n^3 + 4*n^2 - 2*n - 4)*x^2)*P[n - 1, x] - (2*x - (2*n^2 + 6*n + 8)*x^2 + (4*n^2 + 12*n + 8)*x^3)*D[P[n - 1, x], x])/(-n + (n^3 + n^2)*x)]];
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] = (-(n + 1 - (n^3 + 2*n^2 - 3*n + 4)*x - (2*n^3 + 4*n^2 - 2*n - 4)*x^2)*v[i - 1] - (2*x - (2*n^2 + 6*n + 8)*x^2 + (4*n^2 + 12*n + 8)*x^3)*deriv(v[i - 1], x))/(-n + (n^3 + n^2)*x)); 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

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Laguerre’s theorem on multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from T(n,k) = 2^k*binomial(n,2k+1), where 0 <= k <= floor((n-1)/2), n >= 1.?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

References

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.