A199478 → D

nonnegative coefficients

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	3								3
2	5	4							9
3	7	16							23
4	9	36	12						57
5	11	64	60						135
6	13	100	168	32					313
7	15	144	360	192					711
8	17	196	660	640	80				1593
9	19	256	1092	1600	560				3527
10	21	324	1680	3360	2160	192			7737
11	23	400	2448	6272	6160	1536			16839
12	25	484	3420	10752	14560	6720	448		36409
13	27	576	4620	17280	30240	21504	4032		78279
14	29	676	6072	26400	57120	56448	19712	1024	167481

Technical details

Node id: n-36292cabe246c808 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-ff141c83922613d6

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:  5,   4;
  3:  7,  16;
  4:  9,  36,   12;
  5: 11,  64,   60;
  6: 13, 100,  168,   32;
  7: 15, 144,  360,  192;
  8: 17, 196,  660,  640,  80;
  9: 19, 256, 1092, 1600, 560;

%F Formula

The row polynomials satisfy -(2*n^2-n-(2*n^2+8*n)*x)*P(n,x) = -(2*n^2+n+(6*n^2-20*n-16)*x-(8*n^2+8*n-16)*x^2)*P(n-1,x) + ((2*n+1)*x+(14*n+4)*x^2-(16*n+32)*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[(-(2*n^2 + n + (6*n^2 - 20*n - 16)*x - (8*n^2 + 8*n - 16)*x^2)*P[n - 1, x] + ((2*n + 1)*x + (14*n + 4)*x^2 - (16*n + 32)*x^3)*D[P[n - 1, x], x])/(-2*n^2 + n + (2*n^2 + 8*n)*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] = (-(2*n^2 + n + (6*n^2 - 20*n - 16)*x - (8*n^2 + 8*n - 16)*x^2)*v[i - 1] + ((2*n + 1)*x + (14*n + 4)*x^2 - (16*n + 32)*x^3)*deriv(v[i - 1], x))/(-2*n^2 + n + (2*n^2 + 8*n)*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 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 Irregular triangle read by rows: T(n,i) = 2^(i-1)*(binomial(n-i+1,i)+binomial(n-i,i)), n >= 1, 0 <= i <= (n+1)/2.?
  - 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.