A035607 → basis A013609 → basis A013610

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(3 + 6\,x\right)\,P_{n-2}(x) + \left(4 + 6\,x\right)\,P_{n-1}(x)\)

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

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

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	12													19
3	31	96	72												199
4	145	648	936	432											2161
5	673	3936	8424	7776	2592										23401
6	3127	22596	64008	88560	59616	15552									253459
7	14527	124992	440496	812160	824256	435456	93312								2745199
8	67489	673680	2839536	6541344	8880192	7091712	3079296	559872							29733121
9	313537	3561792	17471664	48282048	82114560	87899904	57760128	21275136	3359232						322038001
10	1456615	18554364	103818096	334619424	684039168	918843264	810228096	451816704	144446976	20155392					3487978099
11	6767071	95523744	600384312	2211062400	5279909184	8535995136	9454651776	7079021568	3426416640	967458816	120932352				37778122999
12	31438129	487100184	3397460184	14073322320	38445845184	72684200448	96978321792	91260815616	59324037120	25355483136	6409414656	725594112			409173032881
13	146053729	2464203168	18886737240	86913543456	267329416608	578699313408	903598416000	1024878168576	837614421504	480827031552	183938107392	42084458496	4353564672		4431733435801
14	678529303	12383064372	103447149624	523579325328	1790576396640	4367501425728	7813629712512	10376755547904	10225263702528	7381005327360	3792075761664	1312599748608	274274574336	26121388032	47999891653939

Technical details

Node id: n-42e9b771fcf14435 · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-2ae3ed455d3e70cf

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,      12;
  3:     31,      96,       72;
  4:    145,     648,      936,      432;
  5:    673,    3936,     8424,     7776,     2592;
  6:   3127,   22596,    64008,    88560,    59616,    15552;
  7:  14527,  124992,   440496,   812160,   824256,   435456,    93312;
  8:  67489,  673680,  2839536,  6541344,  8880192,  7091712,  3079296,   559872;
  9: 313537, 3561792, 17471664, 48282048, 82114560, 87899904, 57760128, 21275136, 3359232;

%F Formula

The row polynomials satisfy P(n,x) = (3+6*x)*P(n-2,x) + (4+6*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 + 6*x)*P[n - 2, x] + (4 + 6*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 + 6*x)*v[i - 2] + (4 + 6*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)

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