A330381 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	3	2														5
2	9	11	3													23
3	27	47	25	4												103
4	81	182	143	46	5											457
5	243	668	693	336	75	6										2021
6	729	2369	3052	1981	676	113	7									8927
7	2187	8201	12620	10273	4740	1225	161	8								39415
8	6561	27884	49890	48864	28483	10032	2058	220	9							174001
9	19683	93494	190678	218356	154001	68926	19398	3264	291	10						768101
10	59049	310007	709817	930634	770785	419297	150691	34986	4947	375	11					3390599
11	177147	1018595	2587083	3821574	3636659	2331621	1022385	304414	59697	7227	473	12				14966887
12	531441	3321506	9267001	15229102	16375710	12098922	6272033	2286098	577126	97350	10241	586	13			66067129
13	1594323	10761680	32716835	59205844	70999262	59420694	35587299	15376440	4765982	1038466	152867	14144	715	14		291634565
14	4782969	34676525	114093738	225447231	298327299	279028302	189702009	94907043	34955103	9376302	1788501	232479	19110	861	15	1287337487

Technical details

Node id: n-e453e5c5b6002f20 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-8090db1b43d4bbd1

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:     3,     2;
  2:     9,    11,      3;
  3:    27,    47,     25,      4;
  4:    81,   182,    143,     46,      5;
  5:   243,   668,    693,    336,     75,     6;
  6:   729,  2369,   3052,   1981,    676,   113,     7;
  7:  2187,  8201,  12620,  10273,   4740,  1225,   161,    8;
  8:  6561, 27884,  49890,  48864,  28483, 10032,  2058,  220,   9;
  9: 19683, 93494, 190678, 218356, 154001, 68926, 19398, 3264, 291, 10;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(3 + 3*x + x^2)*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] = 3 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(3 + 3*x + x^2)*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 (−∞, -1]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary), Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary), Geometric basis map (elementary)
Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by 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 Triangle read by rows: T(n,k) is the number of ternary strings of length n with k indispensable digits, with 0 <= k <= …?
  - 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 Triangle, read by rows, of pairwise sums of trinomial coefficients (A027907).?
    
    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 Triangle read by rows: T(n,k) is the number of ternary strings of length n with k indispensable digits, with 0 <= k <= …?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Triangle read by rows: T(n,k) is the number of ternary strings of length n with k indispensable digits, with 0 <= k <= …?
  - ↑ Liu–Wang theorem for three-term recurrences (above)
  - ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

A104029 → x ↦ x/(x+1)

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