A166973 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-6 - 2\,x - x^{2}\right)\,P_{n-2}(x) + \left(-5\,x - 5\,x^{2}\right)\,P_{n-2}'(x) + \left(7 + 2\,x\right)\,P_{n-1}(x) + \left(5\,x\right)\,P_{n-1}'(x)\)

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

\(P_{2}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
1															0
2	1														1
3	7	2													9
4	43	36	3												82
5	259	482	102	4											847
6	1555	5820	2355	220	5										9955
7	9331	67130	46410	7760	405	6									131042
8	55987	757092	843255	224840	20230	672	7								1902083
9	335923	8439986	14627718	5845980	807030	45108	1036	8							30102789
10	2015539	93511692	246703467	142269204	28290255	2366784	89922	1512	9						515248384
11	12093235	1032659690	4087523010	3316591240	913383135	107318610	5998440	164760	2115	10					9475734245
12	72559411	11383443060	66949357695	75098446720	27893700560	4422936672	341148885	13610520	282645	2860	11				186175489039
13	435356467	125362992482	1088264887710	1666333191380	819109272960	170583477504	17441452644	947915760	28313505	459910	3762	12			3888507324096
14	2612138803	1379863630236	17600282692083	36444016869260	23379757586185	6271018930176	826906352272	58797634896	2368826460	54914860	716573	4836	13		85965680296653
15	15672832819	15183724210202	283674318518682	788791397843904	653428718327385	222457295938878	37084817433664	3355738861904	175113356418	5432687260	100534434	1076712	6097	14	2004172331628373

Technical details

Node id: n-a7f62d0ba5f91681 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-a9f5b32ca0d6e2c8

Observed on the stored rows ?

Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -2)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:       0;
   2:       1;
   3:       7,        2;
   4:      43,       36,         3;
   5:     259,      482,       102,         4;
   6:    1555,     5820,      2355,       220,        5;
   7:    9331,    67130,     46410,      7760,      405,       6;
   8:   55987,   757092,    843255,    224840,    20230,     672,     7;
   9:  335923,  8439986,  14627718,   5845980,   807030,   45108,  1036,    8;
  10: 2015539, 93511692, 246703467, 142269204, 28290255, 2366784, 89922, 1512, 9;

%F Formula

The row polynomials satisfy P(n,x) = -(6+2*x+x^2)*P(n-2,x) - (5*x+5*x^2)*d/dx P(n-2,x) + (7+2*x)*P(n-1,x) + 5*x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.

%t Mathematica

P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(6 + 2*x + x^2)*P[n - 2, x] - (5*x + 5*x^2)*D[P[n - 2, x], x] + (7 + 2*x)*P[n - 1, x] + 5*x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = -(6 + 2*x + x^2)*v[i - 2] - (5*x + 5*x^2)*deriv(v[i - 2], x) + (7 + 2*x)*v[i - 1] + 5*x*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle T(n,k) read by rows: T(n, k) = (m*n - m*k + 1)*T(n - 1, k - 1) + (5*k - 4)*(m*k - (m - 1))*T(n - 1, k) where m…?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[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.