A249253 → basis A013609

nonnegative coefficients

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

Recurrence

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

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

\(P_{1}(x) = 4 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	4	2							6
2	17	10							27
3	89	60	4						153
4	551	404	44						999
5	3946	3066	432	8					7452
6	32119	26034	4308	152					62613
7	293017	245264	45336	2240	16				585873
8	2962289	2542912	509736	31168	464				6046569
9	32878196	28803330	6142960	435760	9600	32			68269878
10	397500641	354107450	79311080	6279760	178000	1312			837378243
11	5200386529	4697956572	1094793660	94358560	3195120	36288	64		11090726793
12	73202912047	66920500740	16114637220	1485921760	57469200	865344	3520		157782309831
13	1103244067234	1018906240730	252210265104	24572772280	1053950240	19406688	125440	128	2400006827844
14	17725107987791	16515826176514	4185319880364	426879552680	19892516560	426310752	3741248	9088	38873456174997

Technical details

Node id: n-726576b543a7afe5 · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-c78082d4caa2ada8

Observed on the stored rows ?

Degrees: 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7 (half 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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:        1;
  1:        4,        2;
  2:       17,       10;
  3:       89,       60,       4;
  4:      551,      404,      44;
  5:     3946,     3066,     432,      8;
  6:    32119,    26034,    4308,    152;
  7:   293017,   245264,   45336,   2240,   16;
  8:  2962289,  2542912,  509736,  31168,  464;
  9: 32878196, 28803330, 6142960, 435760, 9600, 32;

%F Formula

The row polynomials satisfy P(n,x) = (1+2*x)*P(n-2,x) + (n+2)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 4 + 2*x;
P[n_, x_] := P[n, x] = Expand[(1 + 2*x)*P[n - 2, x] + (n + 2)*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] = 4 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (1 + 2*x)*v[i - 2] + (n + 2)*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 (−∞, 0]
by Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, 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 ≥ 1
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
  - 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