A269951 → basis A013610

nonnegative coefficients

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

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	1	3							4
3	3	12	9						24
4	11	54	72	27					164
5	49	279	504	351	81				1264
6	261	1659	3645	3537	1539	243			10884
7	1631	11280	28476	34290	21060	6318	729		103784
8	11743	86838	243423	342063	263655	112509	24786	2187	1087204

All 15 rows

Technical details

Node id: n-9796bf969c11399f · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-5da2aa11242d6d62

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:      1;
   2:      1,       3;
   3:      3,      12,        9;
   4:     11,      54,       72,       27;
   5:     49,     279,      504,      351,       81;
   6:    261,    1659,     3645,     3537,     1539,      243;
   7:   1631,   11280,    28476,    34290,    21060,     6318,      729;
   8:  11743,   86838,   243423,   342063,   263655,   112509,    24786,    2187;
   9:  95901,  749091,  2280465,  3603096,  3272724,  1765638,   556227,   94041,   6561;
  10: 876809, 7170747, 23347962, 40477914, 41690943, 26686989, 10685682, 2593782, 347733, 19683;

%F Formula

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

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 1 + 3*x;
P[n_, x_] := P[n, x] = Expand[-(n - 3)*P[n - 2, x] + (n + 3*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 3*x; for(n = 3, N, my(i = n); v[i] = -(n - 3)*v[i - 2] + (n + 3*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
by Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 ≥ 2
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Triangle read by rows, T(n,k) = Sum_{j=0..n} (-1)^(n-j)*C(-j,-n)*S1(j,k), S1 the Stirling cycle numbers A132393, for n>…?
  - 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