A330510 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

\(P_{0}(x) = 2\)

\(P_{1}(x) = 6 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	2															2
1	6	3														9
2	18	18	4													40
3	54	81	37	5												177
4	162	324	226	64	6											782
5	486	1215	1144	501	100	7										3453
6	1458	4374	5194	3098	966	146	8									15244
7	4374	15309	21961	16630	7113	1694	203	9								67293
8	13122	52488	88300	81192	44326	14568	2772	272	10							297050
9	39366	177147	341992	370286	246498	103873	27420	4302	354	11						1311249
10	118098	590490	1286776	1604164	1261710	650614	221080	48348	6402	450	12					5788144
11	354294	1948617	4731301	6676227	6062824	3704477	1545171	436513	80916	9207	561	13				25550121
12	1062882	6377292	17072542	26904200	27719352	19602420	9712846	3377496	811360	129756	12870	688	14			112783718
13	3188646	20726199	60650704	105590683	121737354	97870254	56244456	23286147	6902974	1434862	200772	17563	832	15		497851461
14	9565938	66961566	212642830	405356622	517187584	466086208	305053182	146760054	51904776	13344552	2433574	301366	23478	994	16	2197622740

Technical details

Node id: n-94470298ec1d4aa5 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-db9ee1b7b985cf8f

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:     2;
  1:     6,      3;
  2:    18,     18,      4;
  3:    54,     81,     37,      5;
  4:   162,    324,    226,     64,      6;
  5:   486,   1215,   1144,    501,    100,      7;
  6:  1458,   4374,   5194,   3098,    966,    146,     8;
  7:  4374,  15309,  21961,  16630,   7113,   1694,   203,    9;
  8: 13122,  52488,  88300,  81192,  44326,  14568,  2772,  272,  10;
  9: 39366, 177147, 341992, 370286, 246498, 103873, 27420, 4302, 354, 11;

%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) = 2, P(1,x) = 6 + 3*x.

%t Mathematica

P[0, x_] := 2;
P[1, x_] := 6 + 3*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] = 2; v[2] = 6 + 3*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+1 with k+1 indispensable digits and a nonzer…?
  - 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+1 with k+1 indispensable digits and a nonzer…?
    
    ↑ 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+1 with k+1 indispensable digits and a nonzer…?
  - ↑ Liu–Wang theorem for three-term recurrences (above)
  - ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

A330510 → 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