A131222 → basis A013609

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-4 + 6\,n - 2\,n^{2}\right)\,P_{n-2}(x) + \left(-2 + 2\,x + 3\,n\right)\,P_{n-1}(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	2														3
2	4	10	4													18
3	24	70	48	8												150
4	192	628	572	176	16											1584
5	1920	6868	7540	3240	560	32										20160
6	23040	88608	111496	59880	14800	1632	64									299520
7	322560	1317552	1843240	1166312	367360	58688	4480	128								5080320
8	5160960	22188192	33820720	24315424	9171344	1888768	210560	11776	256							96768000
9	92897280	417460896	683451504	544900096	236770128	58988832	8539776	701184	29952	512						2043740160
10	1857945600	8679600000	15101380224	13122684640	6398690240	1853244960	326770752	35016960	2204160	74240	1024					47377612800
11	40874803200	197640529920	362480716224	338923967008	181986143680	59629984480	12299223552	1612854144	132971520	6617600	180224	2048				1195587993600
12	980995276800	4892015623680	9397321762176	9361385689920	5457664966336	1983677868480	465544004288	71731756800	7261824768	474608640	19137536	430080	4096			32618092953600
13	25505877196800	130789468730880	261790027364736	275689986761280	172612033202752	68569095163712	17945488877888	3159369516160	377046548736	30337122816	1609726976	53647360	1011712	8192		956470394880000
14	714164561510400	3756281629040640	7801215642252288	8630427189925632	5753069832135808	2469080377988224	709708016105088	139885450280576	19134911654144	1819500114432	119092413440	5231163392	146485248	2351104	16384	29994911583436800

Technical details

Node id: n-3bc49e97c8fa21a8 · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-0c5c3044a5453059

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (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:        1,         2;
  2:        4,        10,         4;
  3:       24,        70,        48,         8;
  4:      192,       628,       572,       176,        16;
  5:     1920,      6868,      7540,      3240,       560,       32;
  6:    23040,     88608,    111496,     59880,     14800,     1632,      64;
  7:   322560,   1317552,   1843240,   1166312,    367360,    58688,    4480,    128;
  8:  5160960,  22188192,  33820720,  24315424,   9171344,  1888768,  210560,  11776,   256;
  9: 92897280, 417460896, 683451504, 544900096, 236770128, 58988832, 8539776, 701184, 29952, 512;

%F Formula

The row polynomials satisfy P(n,x) = -(2*n^2-6*n+4)*P(n-2,x) + (3*n-2+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(2*n^2 - 6*n + 4)*P[n - 2, x] + (3*n - 2 + 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] = 1; v[2] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(2*n^2 - 6*n + 4)*v[i - 2] + (3*n - 2 + 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 Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Exponential Riordan array [1, log((1-x)/(1-2x))].?
  - 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