A122542 → basis A013609 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one sign Hadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(1 + 5\,x + 7\,x^{2} + 3\,x^{3}\right)\,P_{n-2}(x) + \left(2 + 6\,x + 4\,x^{2}\right)\,P_{n-1}(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	1	3							4
3	3	14	15						32
4	7	47	101	69					224
5	17	148	470	644	321				1600
6	41	439	1842	3782	3797	1491			11392
7	99	1258	6553	17900	27029	21386	6927		81152
8	239	3515	21859	74479	150109	178913	116753	32181	578048

All 15 rows

Technical details

Node id: n-f45c2041d5d8c4e2 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-44f2cdf251913dc5

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 -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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:    1;
   2:    1,     3;
   3:    3,    14,     15;
   4:    7,    47,    101,      69;
   5:   17,   148,    470,     644,     321;
   6:   41,   439,   1842,    3782,    3797,    1491;
   7:   99,  1258,   6553,   17900,   27029,   21386,    6927;
   8:  239,  3515,  21859,   74479,  150109,  178913,  116753,   32181;
   9:  577,  9640,  69660,  284280,  716422, 1141464, 1122716,  623240,  149505;
  10: 1393, 26059, 214516, 1019660, 3083566, 6151402, 8094020, 6773276, 3270953, 694563;

%F Formula

The row polynomials satisfy (1+x)*P(n,x) = (1+5*x+7*x^2+3*x^3)*P(n-2,x) + (2+6*x+4*x^2)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 3*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 1 + 3*x;
P[n_, x_] := P[n, x] = Expand[Together[((1 + 5*x + 7*x^2 + 3*x^3)*P[n - 2, x] + (2 + 6*x + 4*x^2)*P[n - 1, x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 3*x; for(n = 3, N, my(i = n); v[i] = ((1 + 5*x + 7*x^2 + 3*x^3)*v[i - 2] + (2 + 6*x + 4*x^2)*v[i - 1])/(1 + x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in [-1, 0]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from basis_map_A013609 of Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...]…?
    
    Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...]…?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Liu–Wang theorem with a lag row (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