A084950 → basis A013610

nonnegative coefficients

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

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	3	3							6
2	10	12							22
3	43	60	9						112
4	225	342	81						648
5	1393	2241	675	27					4336
6	9976	16704	5832	432					32944
7	81201	140052	54054	5508	81				280896
8	740785	1307100	542430	67500	2025				2659840
9	7489051	13454655	5898510	842670	36855	243			27721984
10	83120346	151530660	69347340	10964160	609930	8748			315581184
11	1004933203	1854289728	878430555	150108120	9884025	215784	729		3897862144
12	13147251985	24506658162	11943536535	2170411740	161994735	4643730	35721		51934532608
13	185066460993	347962303605	173650811229	33171164145	2728134675	94880079	1148175	2187	742674905088
14	2789144166880	5283382968192	2690225679456	535568483520	47595250080	1913829120	31189536	139968	11347861706752

Technical details

Node id: n-8f6d2c8dc196f9ed · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-6741420c986f1343

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:       3,        3;
  2:      10,       12;
  3:      43,       60,       9;
  4:     225,      342,      81;
  5:    1393,     2241,     675,     27;
  6:    9976,    16704,    5832,    432;
  7:   81201,   140052,   54054,   5508,    81;
  8:  740785,  1307100,  542430,  67500,  2025;
  9: 7489051, 13454655, 5898510, 842670, 36855, 243;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 3*x;
P[n_, x_] := P[n, x] = Expand[(1 + 3*x)*P[n - 2, x] + (n + 1)*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] = 3 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (1 + 3*x)*v[i - 2] + (n + 1)*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 Array of coefficients of denominator polynomials of the n-th approximation of the continued fraction x/(1+x/(2+x/(3+...…?
  - 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