A035607 → basis A013609 → basis A007318

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(3 + 2\,x\right)\,P_{n-2}(x) + \left(4 + 2\,x\right)\,P_{n-1}(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0	1														1
1	1														1
2	7	4													11
3	31	32	8												71
4	145	216	104	16											481
5	673	1312	936	288	32										3241
6	3127	7532	7112	3280	736	64									21851
7	14527	41664	48944	30080	10176	1792	128								147311
8	67489	224560	315504	242272	109632	29184	4224	256							993121
9	313537	1187264	1941296	1788224	1013760	361728	79232	9728	512						6695281
10	1456615	6184788	11535344	12393312	8444928	3781248	1111424	206592	22016	1024					45137291
11	6767071	31841248	66709368	81891200	65184064	35127552	12969344	3236864	522240	49152	2048				304300151
12	31438129	162366728	377495576	521234160	474640064	299111936	133029248	41728768	9041920	1288192	108544	4096			2051487361
13	146053729	821401056	2098526360	3219020128	3300363168	2381478656	1239504000	468622848	127665664	24428544	3115008	237568	8192		13830424921
14	678529303	4127688124	11494127736	19391826864	22105881440	17973256896	10718284928	4744744192	1558491648	374993920	64219136	7409664	516096	16384	93239986331

Technical details

Node id: n-39f1ca0006673cbd · revision 0 · JSON
Definition source: derived:basis_map_A007318
Fingerprint: f6-1bf4f6258e36ae6b

Observed on the stored rows ?

Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
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:      1;
  2:      7,       4;
  3:     31,      32,       8;
  4:    145,     216,     104,      16;
  5:    673,    1312,     936,     288,      32;
  6:   3127,    7532,    7112,    3280,     736,     64;
  7:  14527,   41664,   48944,   30080,   10176,   1792,   128;
  8:  67489,  224560,  315504,  242272,  109632,  29184,  4224,  256;
  9: 313537, 1187264, 1941296, 1788224, 1013760, 361728, 79232, 9728, 512;

%F Formula

The row polynomials satisfy P(n,x) = (3+2*x)*P(n-2,x) + (4+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(3 + 2*x)*P[n - 2, x] + (4 + 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] = 1; for(n = 2, N, my(i = n + 1); v[i] = (3 + 2*x)*v[i - 2] + (4 + 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 Square array A(n,r), the number of neighbors at a sharp Manhattan distance r in a finite n-hypercu…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Square array A(n,r), the number of neighbors at a sharp Manhattan distance r in a finite n-hypercube lattice, read by u…?
    
    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
  - Geometric basis map (elementary) polytool ? step ›
    
    from Square array A(n,r), the number of neighbors at a sharp Manhattan distance r in a finite n-hypercube lattice, read by u…?
    
    ↑ Liu–Wang theorem with a lag row (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)

Related ?

Candidate equalities ?

A035607 → 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