A110552 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(-8 - 24\,x - 18\,x^{2}\right)\,P_{n-2}(x) + \left(6 + 9\,x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 3 + 6\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	3	6														9
2	10	39	36													85
3	36	204	369	216												825
4	136	996	2646	3051	1296											8125
5	528	4704	16344	27864	23409	7776										80625
6	2080	21744	92880	208440	260010	171315	46656									803125
7	8256	98880	499824	1386720	2285820	2242404	1213785	279936								8015625
8	32896	443712	2586528	8530704	17440920	22664124	18299358	8400267	1679616							80078125
9	131328	1969152	12992256	49593600	120857184	195193152	209100528	143362224	57113505	10077696						800390625
10	524800	8657664	63742464	276182784	780663744	1505425824	2007071136	1827737136	1088509266	382972131	60466176					8001953125
11	2098176	37764096	306812160	1486771200	4778663040	10703726208	17058390048	19351275840	15319278900	8062550460	2539638441	362797056				80009765625
12	8390656	163611648	1453579776	7786817280	28031857920	71479698048	132445139904	179743896288	177376110120	124161544980	58532557446	16688841723	2176782336			800048828125
13	33558528	704716800	6795380736	39865706496	158838071040	454085305344	958615637760	1513590997248	1787915855088	1560868198560	979083910728	417946460040	108839648241	13060694016		8000244140625
14	134225920	3020058624	31408386048	200227203072	874516870656	2769289585920	6558814259712	11805817630464	16232939195616	16970742751248	13281409730160	7546234034664	2943023535738	705279071187	78364164096	80001220703125

Technical details

Node id: n-1690f654ed66c83b · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-9d2ce69001c67321

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:      1;
  1:      3,       6;
  2:     10,      39,       36;
  3:     36,     204,      369,      216;
  4:    136,     996,     2646,     3051,      1296;
  5:    528,    4704,    16344,    27864,     23409,      7776;
  6:   2080,   21744,    92880,   208440,    260010,    171315,     46656;
  7:   8256,   98880,   499824,  1386720,   2285820,   2242404,   1213785,    279936;
  8:  32896,  443712,  2586528,  8530704,  17440920,  22664124,  18299358,   8400267,  1679616;
  9: 131328, 1969152, 12992256, 49593600, 120857184, 195193152, 209100528, 143362224, 57113505, 10077696;

%F Formula

The row polynomials satisfy P(n,x) = -(8+24*x+18*x^2)*P(n-2,x) + (6+9*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 6*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 6*x;
P[n_, x_] := P[n, x] = Expand[-(8 + 24*x + 18*x^2)*P[n - 2, x] + (6 + 9*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] = 3 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = -(8 + 24*x + 18*x^2)*v[i - 2] + (6 + 9*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 A triangular array related to A077028 and distributing the values of A007582.?
  - 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