A249247 → basis A013610

nonnegative coefficients

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

Recurrence

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

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

\(P_{1}(x) = 2 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2	3							5
2	5	9							14
3	17	36	9						62
4	73	168	63						304
5	382	927	432	27					1768
6	2365	5949	3159	351					11824
7	16937	43716	25326	3780	81				89840
8	137861	362772	223614	40068	1701				766016
9	1257686	3359475	2169000	440370	26730	243			7253504
10	12714721	34371105	23001930	5114610	389205	7533			75599104
11	141119617	385214688	265268655	63208080	5629095	163296	729		860604160
12	1706150125	4695091524	3309339105	832617360	83282175	3134700	31347		10629646336
13	22321071242	61844763351	44442321084	11683039725	1277921610	57801681	898128	2187	141627819008
14	314201147513	875640228813	639587108853	174323190825	20472036795	1062204759	22009239	124659	2025308051456

Technical details

Node id: n-54cb4eedf261565b · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-706c59cb8c98806f

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:       2,       3;
  2:       5,       9;
  3:      17,      36,       9;
  4:      73,     168,      63;
  5:     382,     927,     432,     27;
  6:    2365,    5949,    3159,    351;
  7:   16937,   43716,   25326,   3780,    81;
  8:  137861,  362772,  223614,  40068,  1701;
  9: 1257686, 3359475, 2169000, 440370, 26730, 243;

%F Formula

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

%t Mathematica

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