A122542 → basis A013609 → reverse

nonnegative coefficients

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

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	2	1														3
3	4	8	3													15
4	8	28	26	7												69
5	16	80	128	80	17											321
6	32	208	480	496	234	41										1491
7	64	512	1552	2288	1748	664	99									6927
8	128	1216	4576	8848	9544	5788	1842	239								32181
9	256	2816	12672	30464	42912	36448	18336	5024	577							149505
10	512	6400	33536	96640	169024	186656	130688	56192	13522	1393						694563
11	1024	14336	85760	288512	604928	827648	750496	446880	167812	36008	3363					3226767
12	2048	31744	213504	822016	2013696	3299840	3698624	2842784	1472456	490876	95050	8119				14990757
13	4096	69632	520192	2256896	6334208	12125440	16257152	15411456	10274736	4709168	1411680	249072	19601			69643329
14	8192	151552	1245184	6012928	19039744	41768704	65378560	74034304	60756576	35755504	14695904	4002480	648634	47321		323545587
15	16384	327680	2936832	15626240	55139328	136542208	244879616	323465472	316661824	228985088	120595888	44929296	11212052	1680184	114243	1503112335

Technical details

Node id: n-695b4df5b709709c · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-c54cc0271404b557

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:   2,    1;
   3:   4,    8,     3;
   4:   8,   28,    26,     7;
   5:  16,   80,   128,    80,     17;
   6:  32,  208,   480,   496,    234,     41;
   7:  64,  512,  1552,  2288,   1748,    664,     99;
   8: 128, 1216,  4576,  8848,   9544,   5788,   1842,   239;
   9: 256, 2816, 12672, 30464,  42912,  36448,  18336,  5024,   577;
  10: 512, 6400, 33536, 96640, 169024, 186656, 130688, 56192, 13522, 1393;

%F Formula

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

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[(2*x + x^2)*P[n - 2, x] + (2 + 2*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] = 2 + x; for(n = 3, N, my(i = n); v[i] = (2*x + x^2)*v[i - 2] + (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, 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 Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from basis_map_A013609 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)

Related ?

Same fingerprint ?

A122542 → basis A013609

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