A116412 → basis A013610 → basis A013610

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(4 + 9\,x\right)\,P_{n-2}(x) + \left(6 + 9\,x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 7 + 9\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	7	9							16
2	46	126	81						253
3	304	1269	1701	729					4003
4	2008	11268	23085	20412	6561				63334
5	13264	93492	258147	348462	229635	59049			1002049
6	87616	743472	2584062	4703508	4723920	2480058	531441		15854077
7	578752	5742720	24069636	55194777	74729790	59698539	26040609	4782969	250837792

All 15 rows

Technical details

Node id: n-3495cde1f187c7d1 · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-cd29e65288cea412

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 0)
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:        7,         9;
  2:       46,       126,         81;
  3:      304,      1269,       1701,        729;
  4:     2008,     11268,      23085,      20412,        6561;
  5:    13264,     93492,     258147,     348462,      229635,       59049;
  6:    87616,    743472,    2584062,    4703508,     4723920,     2480058,      531441;
  7:   578752,   5742720,   24069636,   55194777,    74729790,    59698539,    26040609,    4782969;
  8:  3822976,  43427520,  213129792,  589865976,  1006358985,  1083194856,   717976791,  267846264,   43046721;
  9: 25252864, 323151552, 1817589456, 5894769816, 12142619847, 16467762267, 14698063737, 8322366060, 2711943423, 387420489;

%F Formula

The row polynomials satisfy P(n,x) = (4+9*x)*P(n-2,x) + (6+9*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 7 + 9*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 7 + 9*x;
P[n_, x_] := P[n, x] = Expand[(4 + 9*x)*P[n - 2, x] + (6 + 9*x)*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] = 7 + 9*x; for(n = 2, N, my(i = n + 1); v[i] = (4 + 9*x)*v[i - 2] + (6 + 9*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 - 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 basis_map_A013610 of Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).?
    
    Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
    
    Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Liu–Wang theorem with a lag row (above)
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
    
    ↑ Liu–Wang theorem with a lag row (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Liu–Wang theorem with a lag row (above)
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)
  - Geometric basis map (elementary) polytool ? step ›
    
    from Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Liu–Wang theorem with a lag row (above)
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)

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