A131222 → basis A013610

nonnegative coefficients

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

Recurrence

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

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

\(P_{1}(x) = 1 + 3\,x\)

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	3														4
2	4	15	9													28
3	24	105	108	27												264
4	192	942	1287	594	81											3096
5	1920	10302	16965	10935	2835	243										43200
6	23040	132912	250866	202095	74925	12393	729									696960
7	322560	1976328	4147290	3936303	1859760	445662	51030	2187								12741120
8	5160960	33282288	76096620	82064556	46429929	14342832	2398410	201204	6561							259983360
9	92897280	626191344	1537765884	1839037824	1198648773	447946443	97273386	11980386	767637	19683						5852528640
10	1857945600	13019400000	33978105504	44289060660	32393369340	14073078915	3722123097	598297590	56490210	2854035	59049					143990784000
11	40874803200	296460794880	815581611504	1143868388652	921304852380	452815194645	140095843272	27557125101	3407914620	254402775	10392624	177147				3842231500800
12	980995276800	7338023435520	21143973964896	31594676703480	27629428892076	15063553813770	5302837173843	1225604313450	186112626183	18245550510	1103566761	37200870	531441			110484593049600
13	25505877196800	196184203096320	589027561570656	930453705319320	873848418088932	520696566399438	204410334249693	53980790092515	9663290649441	1166260914819	92824968951	4640365665	131265927	1594323		3405034605772800
14	714164561510400	5634422443560960	17552735195067648	29127691765999008	29124916025187528	18749579120348076	8084017870947018	2390074060653279	490406856886089	69947696781963	6867468673065	452482862013	19005924483	457570701	4782969	111945295016755200

Technical details

Node id: n-e831bdf418cfc910 · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-fd93a149394080ba

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,         3;
  2:        4,        15,          9;
  3:       24,       105,        108,         27;
  4:      192,       942,       1287,        594,         81;
  5:     1920,     10302,      16965,      10935,       2835,       243;
  6:    23040,    132912,     250866,     202095,      74925,     12393,      729;
  7:   322560,   1976328,    4147290,    3936303,    1859760,    445662,    51030,     2187;
  8:  5160960,  33282288,   76096620,   82064556,   46429929,  14342832,  2398410,   201204,   6561;
  9: 92897280, 626191344, 1537765884, 1839037824, 1198648773, 447946443, 97273386, 11980386, 767637, 19683;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 3*x;
P[n_, x_] := P[n, x] = Expand[-(2*n^2 - 6*n + 4)*P[n - 2, x] + (3*n - 2 + 3*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 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = -(2*n^2 - 6*n + 4)*v[i - 2] + (3*n - 2 + 3*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