A122542 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(1 + 6\,x + 9\,x^{2} + 4\,x^{3}\right)\,P_{n-2}(x) + \left(2 + 7\,x + 5\,x^{2}\right)\,P_{n-1}(x)\)

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

\(P_{2}(x) = 1 + 4\,x\)

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	4														5
3	3	18	24													45
4	7	60	162	136												365
5	17	188	750	1274	776											3005
6	41	556	2930	7484	9250	4424										24685
7	99	1590	10398	35394	66066	64074	25224									202845
8	239	4436	34620	147136	367492	539088	429938	143816								1666765
9	577	12152	110164	561116	1755292	3451616	4165174	2819738	819976							13695805
10	1393	32820	338844	2011032	7557816	18644784	30183774	30915204	18177474	4675144						112538285
11	3363	87642	1015020	6876828	30172320	89551188	182035890	250201350	222510330	115616490	26655624					924726045
12	8119	231884	2976766	22665288	113697132	394441968	965466678	1666995360	1989516000	1563007960	727479314	151978696				7598465165
13	19601	608820	8579634	72516902	409337244	1625289696	4653624306	9679909662	14515669440	15302569460	10764777822	4537098042	866515976			62436516605
14	47321	1588124	24372030	226408628	1420189634	6350854392	20826162138	50700037476	91617246702	121362055520	114542986058	72939492300	28088332898	4940494664		513040267885
15	114243	4119678	68386818	692527758	4778662698	23764700478	87824017998	244980075618	518264395758	827410899858	981236552598	838165179882	487485244770	172803625866	28168537224	4215647041245

Technical details

Node id: n-aec4e304d98ffe91 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-549560f24b9e24a2

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,     4;
   3:    3,    18,     24;
   4:    7,    60,    162,     136;
   5:   17,   188,    750,    1274,     776;
   6:   41,   556,   2930,    7484,    9250,     4424;
   7:   99,  1590,  10398,   35394,   66066,    64074,    25224;
   8:  239,  4436,  34620,  147136,  367492,   539088,   429938,   143816;
   9:  577, 12152, 110164,  561116, 1755292,  3451616,  4165174,  2819738,   819976;
  10: 1393, 32820, 338844, 2011032, 7557816, 18644784, 30183774, 30915204, 18177474, 4675144;

%F Formula

The row polynomials satisfy (1+x)*P(n,x) = (1+6*x+9*x^2+4*x^3)*P(n-2,x) + (2+7*x+5*x^2)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 4*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((1 + 6*x + 9*x^2 + 4*x^3)*P[n - 2, x] + (2 + 7*x + 5*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 + 4*x; for(n = 3, N, my(i = n); v[i] = ((1 + 6*x + 9*x^2 + 4*x^3)*v[i - 2] + (2 + 7*x + 5*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_A013610 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