A249253 → basis A013610

nonnegative coefficients

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

Recurrence

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

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

\(P_{1}(x) = 4 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	4	3							7
2	17	15							32
3	89	90	9						188
4	551	606	99						1256
5	3946	4599	972	27					9544
6	32119	39051	9693	513					81376
7	293017	367896	102006	7560	81				770560
8	2962289	3814368	1146906	105192	2349				8031104
9	32878196	43204995	13821660	1470690	48600	243			91424384
10	397500641	531161175	178449930	21194190	901125	9963			1129217024
11	5200386529	7046934858	2463285735	318460140	16175295	275562	729		15045518848
12	73202912047	100380751110	36257933745	5014985940	290937825	6571206	40095		215154131968
13	1103244067234	1528359361095	567473096484	82933106445	5335623090	147369537	1428840	2187	3287494054912
14	17725107987791	24773739264771	9416969730819	1440718490295	100705865085	3237297273	42615153	155277	53460521406464

Technical details

Node id: n-0644de73021829bb · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-728f4a5a60b6910f

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,        3;
  2:       17,       15;
  3:       89,       90,        9;
  4:      551,      606,       99;
  5:     3946,     4599,      972,      27;
  6:    32119,    39051,     9693,     513;
  7:   293017,   367896,   102006,    7560,    81;
  8:  2962289,  3814368,  1146906,  105192,  2349;
  9: 32878196, 43204995, 13821660, 1470690, 48600, 243;

%F Formula

The row polynomials satisfy P(n,x) = (1+3*x)*P(n-2,x) + (n+2)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 3*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 4 + 3*x;
P[n_, x_] := P[n, x] = Expand[(1 + 3*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 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (1 + 3*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