A122542 → basis A013611 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(1 + 7\,x + 11\,x^{2} + 5\,x^{3}\right)\,P_{n-2}(x) + \left(2 + 8\,x + 6\,x^{2}\right)\,P_{n-1}(x)\)

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

\(P_{2}(x) = 1 + 5\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	1	5							6
3	3	22	35						60
4	7	73	237	235					552
5	17	228	1094	2212	1585				5136
6	41	673	4266	13010	19037	10685			47712
7	99	1922	15117	61532	136461	156162	72035		443328
8	239	5357	50275	255737	760541	1321047	1240337	485635	4119168

All 15 rows

Technical details

Node id: n-94a906257cc2f6fd · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-9aec25eefd69f1bb

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,     5;
   3:    3,    22,     35;
   4:    7,    73,    237,     235;
   5:   17,   228,   1094,    2212,     1585;
   6:   41,   673,   4266,   13010,    19037,    10685;
   7:   99,  1922,  15117,   61532,   136461,   156162,    72035;
   8:  239,  5357,  50275,  255737,   760541,  1321047,  1240337,    485635;
   9:  577, 14664, 159836,  974968,  3636742,  8487928, 12098268,   9626312,  3273985;
  10: 1393, 39581, 491268, 3493124, 15669630, 45959286, 88093428, 106375124, 73421337, 22072085;

%F Formula

The row polynomials satisfy (1+x)*P(n,x) = (1+7*x+11*x^2+5*x^3)*P(n-2,x) + (2+8*x+6*x^2)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 5*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 1 + 5*x;
P[n_, x_] := P[n, x] = Expand[Together[((1 + 7*x + 11*x^2 + 5*x^3)*P[n - 2, x] + (2 + 8*x + 6*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 + 5*x; for(n = 3, N, my(i = n); v[i] = ((1 + 7*x + 11*x^2 + 5*x^3)*v[i - 2] + (2 + 8*x + 6*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_A013611 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