A181371 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-6\,n - 27\,n x + 12\,n x^{2} + 6\,n^{2} + 12\,n^{2} x + 3\,n^{3}\right)\,P_{n}(x) = \left(2 - 12\,x + 18\,x^{2} - 8\,x^{3} - 25\,n - 65\,n x + 10\,n x^{2} + 8\,n x^{3} + 25\,n^{2} + 30\,n^{2} x + 8\,n^{2} x^{2} + 7\,n^{3} + 2\,n^{3} x\right)\,P_{n-1}(x) + \left(-15 - 2\,x + 33\,x^{2} - 16\,x^{4} + 20\,n + 16\,n x - 20\,n x^{2} - 16\,n x^{3} + 5\,n^{2} - n^{2} x - 4\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	P_n(1)
0	1				1
1	3				3
2	9	1			10
3	27	6			33
4	80	27	1		108
5	234	108	9		351
6	677	402	54	1	1134
7	1941	1422	270	12	3645

All 15 rows

Technical details

Node id: n-5115618c5fe87f9f · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-103c5766d3e94b1f

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:     9,     1;
  3:    27,     6;
  4:    80,    27,    1;
  5:   234,   108,    9;
  6:   677,   402,   54,   1;
  7:  1941,  1422,  270,  12;
  8:  5523,  4841, 1209,  90,  1;
  9: 15615, 15996, 5013, 540, 15;

%F Formula

The row polynomials satisfy (3*n^3+6*n^2-6*n+(12*n^2-27*n)*x+12*n*x^2)*P(n,x) = (7*n^3+25*n^2-25*n+2+(2*n^3+30*n^2-65*n-12)*x+(8*n^2+10*n+18)*x^2+(8*n-8)*x^3)*P(n-1,x) + (5*n^2+20*n-15-(n^2-16*n+2)*x-(4*n^2+20*n-33)*x^2-16*n*x^3-16*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[((7*n^3 + 25*n^2 - 25*n + 2 + (2*n^3 + 30*n^2 - 65*n - 12)*x + (8*n^2 + 10*n + 18)*x^2 + (8*n - 8)*x^3)*P[n - 1, x] + (5*n^2 + 20*n - 15 - (n^2 - 16*n + 2)*x - (4*n^2 + 20*n - 33)*x^2 - 16*n*x^3 - 16*x^4)*D[P[n - 1, x], x])/(3*n^3 + 6*n^2 - 6*n + (12*n^2 - 27*n)*x + 12*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] = ((7*n^3 + 25*n^2 - 25*n + 2 + (2*n^3 + 30*n^2 - 65*n - 12)*x + (8*n^2 + 10*n + 18)*x^2 + (8*n - 8)*x^3)*v[i - 1] + (5*n^2 + 20*n - 15 - (n^2 - 16*n + 2)*x - (4*n^2 + 20*n - 33)*x^2 - 16*n*x^3 - 16*x^4)*deriv(v[i - 1], x))/(3*n^3 + 6*n^2 - 6*n + (12*n^2 - 27*n)*x + 12*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

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 Triangle read by rows: T(n,k) is the number of ternary words (i.e., finite sequences of 0's, 1's and 2's) of length n h…?
  - 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.