A249253 → basis A013611

nonnegative coefficients

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

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	4	4							8
2	17	20							37
3	89	120	16						225
4	551	808	176						1535
5	3946	6132	1728	64					11870
6	32119	52068	17232	1216					102635
7	293017	490528	181344	17920	256				983065
8	2962289	5085824	2038944	249344	7424				10343825
9	32878196	57606660	24571840	3486080	153600	1024			118697400
10	397500641	708214900	317244320	50238080	2848000	41984			1476087925
11	5200386529	9395913144	4379174640	754868480	51121920	1161216	4096		19782630025
12	73202912047	133841001480	64458548880	11887374080	919507200	27691008	225280		284337259975
13	1103244067234	2037812481460	1008841060416	196582178240	16863203840	621014016	8028160	16384	4363972049750
14	17725107987791	33031652353028	16741279521456	3415036421440	318280264960	13641944064	239439872	1163264	71245239095875

Technical details

Node id: n-67fb46a4c660ceb5 · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-001856914dacfa6c

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,        4;
  2:       17,       20;
  3:       89,      120,       16;
  4:      551,      808,      176;
  5:     3946,     6132,     1728,      64;
  6:    32119,    52068,    17232,    1216;
  7:   293017,   490528,   181344,   17920,    256;
  8:  2962289,  5085824,  2038944,  249344,   7424;
  9: 32878196, 57606660, 24571840, 3486080, 153600, 1024;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 4 + 4*x;
P[n_, x_] := P[n, x] = Expand[(1 + 4*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 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (1 + 4*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