A144440 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

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

\(P_{0}(x) = 1\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	2														3
2	1	13	13													27
3	1	57	165	110												333
4	1	233	1482	2498	1249											5463
5	1	937	11713	37502	44540	17816										112509
6	1	3753	87378	473869	1003482	919857	306619									2794959
7	1	15017	633225	5475664	18405470	29012586	21650398	6185828								81378189
8	1	60073	4518922	60095833	301187155	726310009	910341541	572775604	143193901							2718483039
9	1	240297	31977873	639054432	4594813845	15909223716	29786013600	30959883060	16837195095	3741598910						102500000829
10	1	961193	225226818	6660104289	66956023002	319958822250	839473553064	1281747653970	1137456539190	544679750465	108935950093					4306094584335
11	1	3844777	1582114585	68499511944	945702543120	6072427082859	21450809599149	45125005160568	58098847211040	44968294266425	19234463598383	3497175199706				199462810132557
12	1	15379113	11096909562	698330600153	13068283685265	110619517311297	511957387328190	1425244602641418	2496709962590166	2776939879526420	1905395314145163	736367639039934	122727939839989			10099739968996671
13	1	61516457	77766796833	7076839156112	177779569389715	1955910854739654	11628776542405596	41677098714721020	95398102369785915	142978447211198900	139907712669198628	86201758053597396	30381996056871550	4674153239518700		554988889948896477
14	1	246065833	544721297458	71423862801809	2391107915312672	33824787490146728	254568302239270128	1151339433285892821	3349337435653140159	6495513047223697742	8499335011396275326	7421953584817682309	4149120674839711501	1344059266478506669	192008466639786667	32893523086809587823

Technical details

Node id: n-b0c8b82f34ff32ea · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-4eb074acc1a49a5d

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 0)
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: 1;
  1: 1,      2;
  2: 1,     13,       13;
  3: 1,     57,      165,       110;
  4: 1,    233,     1482,      2498,       1249;
  5: 1,    937,    11713,     37502,      44540,       17816;
  6: 1,   3753,    87378,    473869,    1003482,      919857,      306619;
  7: 1,  15017,   633225,   5475664,   18405470,    29012586,    21650398,     6185828;
  8: 1,  60073,  4518922,  60095833,  301187155,   726310009,   910341541,   572775604,   143193901;
  9: 1, 240297, 31977873, 639054432, 4594813845, 15909223716, 29786013600, 30959883060, 16837195095, 3741598910;

%F Formula

The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (3*x+9*x^2+9*x^3+3*x^4)*P(n-2,x) + (1+4*x+5*x^2+2*x^3)*P(n-1,x) + (3*x+9*x^2+9*x^3+3*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((3*x + 9*x^2 + 9*x^3 + 3*x^4)*P[n - 2, x] + (1 + 4*x + 5*x^2 + 2*x^3)*P[n - 1, x] + (3*x + 9*x^2 + 9*x^3 + 3*x^4)*D[P[n - 1, x], x])/(1 + 2*x + x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((3*x + 9*x^2 + 9*x^3 + 3*x^4)*v[i - 2] + (1 + 4*x + 5*x^2 + 2*x^3)*v[i - 1] + (3*x + 9*x^2 + 9*x^3 + 3*x^4)*deriv(v[i - 1], x))/(1 + 2*x + x^2)); 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, 0]
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 Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, 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 Triangle T(n,k) by rows: T(n, k) = (3*n-3*k+1)*T(n-1, k-1) +(3*k-2)*T(n-1, k) + 3*T(n-2, k-1) with T(n, 1) = T(n, n) = …?
  - Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
  - 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) by rows: T(n, k) = (3*n-3*k+1)*T(n-1, k-1) +(3*k-2)*T(n-1, k) + 3*T(n-2, k-1) with T(n, 1) = T(n, n) = …?
    
    ↑ Liu–Wang theorem with a lag row (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)

Related ?

Same fingerprint ?

A144440 → x ↦ x+1

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