A199881 → basis A007318

nonnegative coefficients

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

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	P_n(1)
0	2	1								3
1	2	3	1							6
2	4	8	5	1						18
3	6	17	17	7	1					48
4	10	35	47	30	9	1				132
5	16	68	116	101	47	11	1			360
6	26	129	266	294	187	68	13	1		984
7	42	239	579	777	629	313	93	15	1	2688

All 15 rows

Technical details

Node id: n-fbb0841efccb28a7 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-1952f04fdce2e644

Observed on the stored rows ?

Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (affine: slope 1, intercept 1)
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;
  1:   2,   3,    1;
  2:   4,   8,    5,    1;
  3:   6,  17,   17,    7,    1;
  4:  10,  35,   47,   30,    9,    1;
  5:  16,  68,  116,  101,   47,   11,    1;
  6:  26, 129,  266,  294,  187,   68,   13,   1;
  7:  42, 239,  579,  777,  629,  313,   93,  15,   1;
  8:  68, 436, 1213, 1916, 1887, 1197,  487, 122,  17,  1;
  9: 110, 785, 2467, 4485, 5209, 4026, 2090, 717, 155, 19, 1;

%F Formula

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

%t Mathematica

P[0, x_] := 2 + x;
P[1, x_] := 2 + 3*x + x^2;
P[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 2, x] + (1 + 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 + x; v[2] = 2 + 3*x + x^2; for(n = 2, N, my(i = n + 1); v[i] = (1 + x)*v[i - 2] + (1 + 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 Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (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 T(n,k), read by rows, given by (1,-1,0,0,0,0,0,0,0,0,0,...) DELTA (1,0,-1,1,0,0,0,0,0,0,0,...) where DELTA is …?
  - 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 T(n,k), read by rows, given by (1,-1,0,0,0,0,0,0,0,0,0,...) DELTA (1,0,-1,1,0,0,0,0,0,0,0,...) where DELTA is …?
  - ↑ 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