A306364 → D+1

nonnegative coefficients

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

Recurrence

\(\left(10 - 14\,x + 4\,x^{2} - 3\,n + 4\,n x + n^{2}\right)\,P_{n}(x) = \left(16 - 32\,x + 18\,x^{2} - 4\,x^{3} + 6\,n + n x - 10\,n x^{2} + 4\,n x^{3} + n^{2} + 4\,n^{2} x^{2} + n^{3} + n^{3} x\right)\,P_{n-1}(x) + \left(8\,x - 20\,x^{2} + 20\,x^{3} - 8\,x^{4} - 2\,n x + 10\,n x^{2} - 8\,n x^{3} + 2\,n^{2} x - 2\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	P_n(1)
0	1				1
1	3				3
2	15	3			18
3	105	45			150
4	900	630	45		1575
5	8820	9450	1575		19845
6	95760	151200	42525	1575	291060
7	1134000	2540160	1091475	99225	4864860

All 15 rows

Technical details

Node id: n-679ddd982dbce61c · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-eac7d4e10a398dd8

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:         3;
  2:        15,         3;
  3:       105,        45;
  4:       900,       630,        45;
  5:      8820,      9450,      1575;
  6:     95760,    151200,     42525,      1575;
  7:   1134000,   2540160,   1091475,     99225;
  8:  14515200,  44452800,  27783000,   4365900,   99225;
  9: 199584000, 808315200, 707275800, 170270100, 9823275;

%F Formula

The row polynomials satisfy (n^2-3*n+10+(4*n-14)*x+4*x^2)*P(n,x) = (n^3+n^2+6*n+16+(n^3+n-32)*x+(4*n^2-10*n+18)*x^2+(4*n-4)*x^3)*P(n-1,x) + ((2*n^2-2*n+8)*x-(2*n^2-10*n+20)*x^2-(8*n-20)*x^3-8*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[((n^3 + n^2 + 6*n + 16 + (n^3 + n - 32)*x + (4*n^2 - 10*n + 18)*x^2 + (4*n - 4)*x^3)*P[n - 1, x] + ((2*n^2 - 2*n + 8)*x - (2*n^2 - 10*n + 20)*x^2 - (8*n - 20)*x^3 - 8*x^4)*D[P[n - 1, x], x])/(n^2 - 3*n + 10 + (4*n - 14)*x + 4*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] = ((n^3 + n^2 + 6*n + 16 + (n^3 + n - 32)*x + (4*n^2 - 10*n + 18)*x^2 + (4*n - 4)*x^3)*v[i - 1] + ((2*n^2 - 2*n + 8)*x - (2*n^2 - 10*n + 20)*x^2 - (8*n - 20)*x^3 - 8*x^4)*deriv(v[i - 1], x))/(n^2 - 3*n + 10 + (4*n - 14)*x + 4*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

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

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangular array of the number of binary, rooted, leaf-labeled tree topologies with n leaves and k cherries, n >= 2, 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

[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.