A111006 → basis A008288

nonnegative coefficients

Proved for all n all roots real, in [−3−2√2, −3+2√2]coefficients of one sign Hadamard kernel

Recurrence

\(P_{n}(x) = \left(2\,x\right)\,P_{n-4}(x) + \left(3\,x + x^{2}\right)\,P_{n-3}(x) + \left(2 + 6\,x + x^{2}\right)\,P_{n-2}(x) + \left(1 + x\right)\,P_{n-1}(x)\)

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

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

\(P_{2}(x) = 3 + 7\,x + 2\,x^{2}\)

\(P_{3}(x) = 5 + 21\,x + 17\,x^{2} + 3\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	1														2
2	3	7	2													12
3	5	21	17	3												46
4	11	63	91	40	5											210
5	21	157	345	273	82	8										886
6	43	391	1155	1383	722	163	13									3870
7	85	917	3441	5681	4469	1721	311	21								16646
8	171	2127	9675	20583	22291	12760	3864	581	34							72086
9	341	4813	25801	67729	95189	74701	33268	8276	1065	55						311238
10	683	10775	66515	208423	363923	370271	224506	81330	17133	1926	89					1345574
11	1365	23813	166497	608049	1277685	1618853	1268721	621671	189166	34518	3444	144				5813926
12	2731	52191	407707	1702823	4200787	6426919	6266947	3952592	1616847	423203	68059	6103	233			25127142
13	5461	113469	979769	4611345	13101013	23630141	27846249	21804385	11431566	4000612	917445	131825	10733	377		108584390
14	10923	245159	2318883	12151527	39146067	81661487	113647371	107458447	69748538	31155693	9506812	1938061	251577	18755	610	469259910

Technical details

Node id: n-ff2c124f57c19690 · revision 0 · JSON
Definition source: derived:basis_map_A008288
Fingerprint: f6-ce6d9281d03644ef

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): no

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0:   1;
  1:   1,    1;
  2:   3,    7,     2;
  3:   5,   21,    17,     3;
  4:  11,   63,    91,    40,     5;
  5:  21,  157,   345,   273,    82,     8;
  6:  43,  391,  1155,  1383,   722,   163,    13;
  7:  85,  917,  3441,  5681,  4469,  1721,   311,   21;
  8: 171, 2127,  9675, 20583, 22291, 12760,  3864,  581,   34;
  9: 341, 4813, 25801, 67729, 95189, 74701, 33268, 8276, 1065, 55;

%F Formula

The row polynomials satisfy P(n,x) = 2*x*P(n-4,x) + (3*x+x^2)*P(n-3,x) + (2+6*x+x^2)*P(n-2,x) + (1+x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + x, P(2,x) = 3 + 7*x + 2*x^2, P(3,x) = 5 + 21*x + 17*x^2 + 3*x^3.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + x;
P[2, x_] := 3 + 7*x + 2*x^2;
P[3, x_] := 5 + 21*x + 17*x^2 + 3*x^3;
P[n_, x_] := P[n, x] = Expand[2*x*P[n - 4, x] + (3*x + x^2)*P[n - 3, x] + (2 + 6*x + x^2)*P[n - 2, x] + (1 + 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] = 1 + x; v[3] = 3 + 7*x + 2*x^2; v[4] = 5 + 21*x + 17*x^2 + 3*x^3; for(n = 4, N, my(i = n + 1); v[i] = 2*x*v[i - 4] + (3*x + x^2)*v[i - 3] + (2 + 6*x + x^2)*v[i - 2] + (1 + 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 [−3−2√2, −3+2√2] (rational outer bound [-1564556493/268435456, -46056243/268435456])
by Affine-current basis map theorem
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine-current basis map theorem
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine-current basis map theorem

Proof chains ? JSON

Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ? step ›
- from Another version of Fibonacci-Pascal triangle A037027.?
  - 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