A121574 → basis A013609 → basis A007318

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(3 + 2\,x\right)\,P_{n-2}(x) + \left(5 + 2\,x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 5 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	5	2							7
2	28	22	4						54
3	155	182	68	8					413
4	859	1342	760	184	16				3161
5	4760	9284	7052	2600	464	32			24192
6	26377	61684	58792	29176	7936	1120	64		185149
7	146165	398546	457052	285368	104624	22496	2624	128	1417003

All 15 rows

Technical details

Node id: n-1af6da463398282c · revision 0 · JSON
Definition source: derived:basis_map_A007318
Fingerprint: f6-7e54cdbe5f0f29a0

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:       5,        2;
  2:      28,       22,        4;
  3:     155,      182,       68,        8;
  4:     859,     1342,      760,      184,       16;
  5:    4760,     9284,     7052,     2600,      464,      32;
  6:   26377,    61684,    58792,    29176,     7936,    1120,      64;
  7:  146165,   398546,   457052,   285368,   104624,   22496,    2624,    128;
  8:  809956,  2522866,  3382096,  2546056,  1176016,  340960,   60544,   6016,   256;
  9: 4488275, 15722210, 24124460, 21264680, 11856800, 4333568, 1037504, 156800, 13568, 512;

%F Formula

The row polynomials satisfy P(n,x) = (3+2*x)*P(n-2,x) + (5+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 5 + 2*x;
P[n_, x_] := P[n, x] = Expand[(3 + 2*x)*P[n - 2, x] + (5 + 2*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] = 5 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (3 + 2*x)*v[i - 2] + (5 + 2*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_A013609 of Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).?
    
    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 reversal of Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).?
    
    ↑ 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/(1-2*x), x*(1+x)/(1-2*x)).?
    
    ↑ Liu–Wang theorem with a lag row (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
  - Geometric basis map (elementary) polytool ? step ›
    
    from Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).?
    
    ↑ Liu–Wang theorem with a lag row (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Candidate equalities ?

A121574 → basis A013609 → x ↦ x+1 15 rows; no all-index equality certificate was found

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