A122542 → x ↦ x+1 → basis A013609

nonnegative coefficients

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

Recurrence

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

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

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

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	2														4
3	8	12	4													24
4	28	60	40	8												136
5	100	276	272	112	16											776
6	356	1204	1568	976	288	32										4424
7	1268	5076	8208	6832	3072	704	64									25224
8	4516	20884	40320	42000	25408	8896	1664	128								143816
9	16084	84372	189296	236720	180032	85056	24320	3840	256							819976
10	57284	336084	859040	1253392	1148352	683840	264192	63744	8704	512						4675144
11	204020	1323732	3796624	6330288	6785344	4878400	2379008	775936	161792	19456	1024					26655624
12	726628	5165972	16427584	30808976	37820096	31870272	18789888	7741696	2182144	400384	43008	2048				151978696
13	2587924	20006676	69855408	145535920	201309504	194578496	134625024	67114752	23905280	5927936	970752	94208	4096			866515976
14	9217028	76981076	292766688	670791696	1032258496	1125735232	894352384	523657472	225793024	70759424	15654912	2314240	204800	8192		4940494664
15	32826932	294566484	1211986384	3028691120	5132049728	6233498688	5592934656	3763156736	1906734080	723530752	202280960	40382464	5439488	442368	16384	28168537224

Technical details

Node id: n-314bd31589a6fac5 · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-9e16e8dbf47e29d6

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,      2;
   3:     8,     12,      4;
   4:    28,     60,     40,       8;
   5:   100,    276,    272,     112,      16;
   6:   356,   1204,   1568,     976,     288,     32;
   7:  1268,   5076,   8208,    6832,    3072,    704,     64;
   8:  4516,  20884,  40320,   42000,   25408,   8896,   1664,   128;
   9: 16084,  84372, 189296,  236720,  180032,  85056,  24320,  3840,  256;
  10: 57284, 336084, 859040, 1253392, 1148352, 683840, 264192, 63744, 8704, 512;

%F Formula

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

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[(2 + 2*x)*P[n - 2, x] + (3 + 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 + 2*x; for(n = 3, N, my(i = n); v[i] = (2 + 2*x)*v[i - 2] + (3 + 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 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 shift_x_plus_one 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,…?
  - Affine, reversal and Möbius substitutions (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
  - Affine, reversal and Möbius substitutions (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)
  - 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)
  - 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