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	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

All 15 rows

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