A122542 → x ↦ x+1 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	1	2														3
3	3	10	8													21
4	7	34	54	28												123
5	17	108	254	262	100											741
6	41	322	1002	1544	1178	356										4443
7	99	926	3582	7334	8382	5070	1268									26661
8	239	2594	11992	30608	46580	42264	21170	4516								159963
9	577	7128	38324	117124	222532	269152	202374	86486	16084							959781
10	1393	19298	118284	420992	958824	1449120	1453326	932648	347514	57284						5758683
11	3363	51634	355324	1443188	3831176	6945708	8708906	7457186	4173262	1378334	204020					34552101
12	8119	136834	1044514	4766788	14450284	30552520	45973270	49231956	36770192	18241576	5409922	726628				207312603
13	19601	359748	3016482	15279546	52072320	125781504	220812482	283859042	265198884	175606220	78230118	21051750	2587924			1243875621
14	47321	939490	8583462	47782536	180822954	491216916	985500874	1478412336	1658387494	1373825156	816953930	330233400	81330826	9217028		7463253723
15	114243	2439494	24120322	146363946	608937438	1837460250	4146954130	7111186790	9310412970	9261733750	6890829874	3718283134	1375565262	312293806	32826932	44779522341

Technical details

Node id: n-bd5fbbc3114b48df · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-25a7cba39871cb33

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,     2;
   3:    3,    10,      8;
   4:    7,    34,     54,     28;
   5:   17,   108,    254,    262,    100;
   6:   41,   322,   1002,   1544,   1178,     356;
   7:   99,   926,   3582,   7334,   8382,    5070,    1268;
   8:  239,  2594,  11992,  30608,  46580,   42264,   21170,   4516;
   9:  577,  7128,  38324, 117124, 222532,  269152,  202374,  86486,  16084;
  10: 1393, 19298, 118284, 420992, 958824, 1449120, 1453326, 932648, 347514, 57284;

%F Formula

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

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((1 + 4*x + 5*x^2 + 2*x^3)*P[n - 2, x] + (2 + 5*x + 3*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 + 2*x; for(n = 3, N, my(i = n); v[i] = ((1 + 4*x + 5*x^2 + 2*x^3)*v[i - 2] + (2 + 5*x + 3*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 shift_x_plus_one 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,…?
  - Affine, reversal and Möbius substitutions (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
  - Affine, reversal and Möbius substitutions (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)
  - 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)
  - 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