A249253 → D+1

nonnegative coefficients

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

Recurrence

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

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

\(P_{1}(x) = 4 + x\)

\(P_{2}(x) = 17 + 5\,x\)

\(P_{3}(x) = 88 + 30\,x + x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	4	1							5
2	17	5							22
3	88	30	1						119
4	540	202	11						753
5	3840	1530	108	1					5479
6	31080	12960	1077	19					45136
7	282240	121800	11328	280	1				415649
8	2842560	1260000	127260	3896	29				4233745
9	31449600	14243040	1532160	54460	600	1			47279861
10	379209600	174787200	19761840	784560	11125	41			574554366
11	4949683200	2315174400	272522880	11783520	199680	1134	1		7549364815
12	69535065600	32931360000	4007646720	185472000	3591000	27042	55		106663162417
13	1046139494400	500756256000	62669376000	3065610240	65842560	606438	1960	1	1612697187599
14	16781821056000	8107581081600	1039134096000	53229052800	1242410400	13320720	58457	71	25983021076048

Technical details

Node id: n-083ccef26f07c0ed · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-089c7f24353cb775

Observed on the stored rows ?

Degrees: 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7 (half affine: slope 1, intercept 1)
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:        4,        1;
  2:       17,        5;
  3:       88,       30,       1;
  4:      540,      202,      11;
  5:     3840,     1530,     108,     1;
  6:    31080,    12960,    1077,    19;
  7:   282240,   121800,   11328,   280,   1;
  8:  2842560,  1260000,  127260,  3896,  29;
  9: 31449600, 14243040, 1532160, 54460, 600, 1;

%F Formula

The row polynomials satisfy P(n,x) = -x^2*P(n-4,x) - 2*n*x*P(n-3,x) - (n^2+n-2*x)*P(n-2,x) + (2*n+2)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 4 + x, P(2,x) = 17 + 5*x, P(3,x) = 88 + 30*x + x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 4 + x;
P[2, x_] := 17 + 5*x;
P[3, x_] := 88 + 30*x + x^2;
P[n_, x_] := P[n, x] = Expand[-x^2*P[n - 4, x] - 2*n*x*P[n - 3, x] - (n^2 + n - 2*x)*P[n - 2, x] + (2*n + 2)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + x; v[3] = 17 + 5*x; v[4] = 88 + 30*x + x^2; for(n = 4, N, my(i = n + 1); v[i] = -x^2*v[i - 4] - 2*n*x*v[i - 3] - (n^2 + n - 2*x)*v[i - 2] + (2*n + 2)*v[i - 1]); 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: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
  - 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.