A122542 → basis A013611 → basis A013609

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(5 + 8\,x\right)\,P_{n-2}(x) + \left(6 + 8\,x\right)\,P_{n-1}(x)\)

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

\(P_{2}(x) = 5 + 8\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	5	8														13
3	35	96	64													195
4	235	936	1216	512												2899
5	1585	8256	15872	13312	4096											43121
6	10685	68776	174848	219136	135168	32768										641381
7	72035	552096	1744704	2907136	2691072	1310720	262144									9539907
8	485635	4318216	16309440	33894912	41832448	30638080	12320768	2097152								141896651
9	3273985	33131136	145542656	362338304	558866432	546570240	330825728	113246208	16777216							2110571905
10	22072085	250454856	1254397952	3638321152	6732226560	8238202880	6664224768	3435134976	1023410176	134217728						31392663133
11	148802435	1871153376	10522788928	34841143296	75192967168	110490812416	111917662208	77137444864	34611396608	9126805504	1073741824					466934718627
12	1003175035	13846190696	86381589184	321455960576	792654651392	1359537438720	1654659219456	1428655439872	857366069248	340510375936	80530636800	8589934592				6945190681507
13	6763062385	101648730816	696642232320	2878176504832	7982269575168	15652459642880	22247769636864	23090234916864	17363596476416	9224516009984	3285649981440	704374636544	68719476736			103302820883249
14	45594249485	741253237736	5535720711936	25140529403904	77453950431232	170911838863360	277855890571264	336904117551104	304619532058624	202817348304896	96636764160000	31198642438144	6116033429504	549755813888		1536526971225077
15	307380808835	5374621575456	43440751181632	215092962501632	728784697733120	1787223091445760	3270888579661824	4537705161555968	4794489994477568	3838891698159616	2292583749386240	990093040943104	292263934558208	52776558133248	4398046511104	22854314268633315

Technical details

Node id: n-305613286c61eb36 · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-8784a47e85079371

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 -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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:        1;
   2:        5,         8;
   3:       35,        96,         64;
   4:      235,       936,       1216,        512;
   5:     1585,      8256,      15872,      13312,       4096;
   6:    10685,     68776,     174848,     219136,     135168,      32768;
   7:    72035,    552096,    1744704,    2907136,    2691072,    1310720,     262144;
   8:   485635,   4318216,   16309440,   33894912,   41832448,   30638080,   12320768,    2097152;
   9:  3273985,  33131136,  145542656,  362338304,  558866432,  546570240,  330825728,  113246208,   16777216;
  10: 22072085, 250454856, 1254397952, 3638321152, 6732226560, 8238202880, 6664224768, 3435134976, 1023410176, 134217728;

%F Formula

The row polynomials satisfy P(n,x) = (5+8*x)*P(n-2,x) + (6+8*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 5 + 8*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 5 + 8*x;
P[n_, x_] := P[n, x] = Expand[(5 + 8*x)*P[n - 2, x] + (6 + 8*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 5 + 8*x; for(n = 3, N, my(i = n); v[i] = (5 + 8*x)*v[i - 2] + (6 + 8*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, 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 Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 ≥ 2
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from basis_map_A013611 of Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...]…?
    
    Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...]…?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Liu–Wang theorem with a lag row (above)

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