A143491 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-4 - 4\,x - x^{2} - 4\,n - 2\,n x - n^{2}\right)\,P_{n-2}(x) + \left(5 + 2\,x + 2\,n\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 5 + 2\,x\)

\(P_{1}(x) = 26 + 18\,x + 3\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	P_n(1)
0	5	2															7
1	26	18	3														47
2	154	142	42	4													342
3	1044	1160	465	80	5												2754
4	8028	10208	4995	1180	135	6											24552
5	69264	97720	55272	16100	2555	210	7										241128
6	663696	1018008	644028	218596	43120	4956	308	8									2592720
7	6999840	11507472	7967292	3044664	704445	101304	8862	432	9								30334320
8	80627040	140581872	104901420	44114360	11554725	1959678	215670	14880	585	10							383970240
9	1007441280	1848236544	1469689848	669240880	193799650	37381806	4874331	425040	23760	770	11						5231113920
10	13575738240	26041957632	21878322840	10659717904	3355946550	718523058	106978410	11096184	787050	36410	990	12					76349105280
11	196287356160	391738883328	345359456208	178407147776	60307899080	14086458672	2335974641	277607616	23501907	1384240	53911	1248	13				1188825724800
12	3031488633600	6268657962240	5768000168112	3136586491584	1127627420920	283666358976	51473821399	6833799544	664837173	46876830	2331329	77532	1547	14			19675048780800
13	49811492505600	106361504663040	101690989633152	57876050756160	21962771849200	5891814648720	1154525227856	168168017160	18325419255	1488737250	88865777	3783780	108745	1890	15		344937224217600
14	867718162483200	1907768564282880	1888289080758144	1119480849032256	445712184881600	126516175247280	26500712630392	4178313694984	500721146640	45670110200	3148329184	161268744	5947760	149240	2280	16	6386713749964800

Technical details

Node id: n-f6e156a1eb27b6e0 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-6443481be5042dac

Observed on the stored rows ?

Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (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:          5,          2;
  1:         26,         18,          3;
  2:        154,        142,         42,         4;
  3:       1044,       1160,        465,        80,         5;
  4:       8028,      10208,       4995,      1180,       135,        6;
  5:      69264,      97720,      55272,     16100,      2555,      210,       7;
  6:     663696,    1018008,     644028,    218596,     43120,     4956,     308,      8;
  7:    6999840,   11507472,    7967292,   3044664,    704445,   101304,    8862,    432,     9;
  8:   80627040,  140581872,  104901420,  44114360,  11554725,  1959678,  215670,  14880,   585,  10;
  9: 1007441280, 1848236544, 1469689848, 669240880, 193799650, 37381806, 4874331, 425040, 23760, 770, 11;

%F Formula

The row polynomials satisfy P(n,x) = -(n^2+4*n+4+(2*n+4)*x+x^2)*P(n-2,x) + (2*n+5+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 5 + 2*x, P(1,x) = 26 + 18*x + 3*x^2.

%t Mathematica

P[0, x_] := 5 + 2*x;
P[1, x_] := 26 + 18*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[-(n^2 + 4*n + 4 + (2*n + 4)*x + x^2)*P[n - 2, x] + (2*n + 5 + 2*x)*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] = 5 + 2*x; v[2] = 26 + 18*x + 3*x^2; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 + 4*n + 4 + (2*n + 4)*x + x^2)*v[i - 2] + (2*n + 5 + 2*x)*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
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Unsigned 2-Stirling numbers of the first kind.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Parent-row provers of polytool polytool ? step › direct check
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

Related ?

Same fingerprint ?

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.