A154227 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

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

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

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

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	10	10													21
3	1	22	60	40												123
4	1	39	269	460	230											999
5	1	62	830	2720	3250	1300										8163
6	1	92	2074	13004	29102	27120	9040									80433
7	1	130	4526	49264	182910	300244	227080	64880								829035
8	1	177	8971	155786	959948	2560834	3357558	2146240	536560							9726075
9	1	234	16530	429808	4229970	17250300	35052696	37864176	20886840	4641520						120372075
10	1	302	28746	1066636	15963548	99348684	301924908	498583440	459865860	223480000	44696000					1644958125
11	1	382	47680	2431720	52911184	494738464	2176083336	5182066944	7144548756	5725790968	2482868080	451430560				23712918075
12	1	475	76017	5170448	157454386	2150307576	13658973376	46050130488	90354942312	107233774780	76118923276	29821182720	4970197120			370521132975
13	1	582	117182	10368992	428132282	8276221268	75462801568	353804366240	955060982688	1582238307904	1641972143856	1043968229312	372713919760	57340603040		6091276194675
14	1	704	175466	19782396	1078446066	28647465912	368946679544	2401843683616	8827762178848	19860969009280	28517094810704	26330236283744	15173463097904	4977734023360	711104860480	107198900498025

Technical details

Node id: n-5bcd698d4270d7b8 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-d769fcdab31c02d6

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,  10,    10;
  3: 1,  22,    60,     40;
  4: 1,  39,   269,    460,     230;
  5: 1,  62,   830,   2720,    3250,     1300;
  6: 1,  92,  2074,  13004,   29102,    27120,     9040;
  7: 1, 130,  4526,  49264,  182910,   300244,   227080,    64880;
  8: 1, 177,  8971, 155786,  959948,  2560834,  3357558,  2146240,   536560;
  9: 1, 234, 16530, 429808, 4229970, 17250300, 35052696, 37864176, 20886840, 4641520;

%F Formula

The row polynomials satisfy (2+2*x)*P(n,x) = ((n^2+3*n+2)*x+(2*n^2+6*n+4)*x^2+(n^2+3*n+2)*x^3)*P(n-2,x) + (2+6*x+4*x^2)*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[(((n^2 + 3*n + 2)*x + (2*n^2 + 6*n + 4)*x^2 + (n^2 + 3*n + 2)*x^3)*P[n - 2, x] + (2 + 6*x + 4*x^2)*P[n - 1, x])/(2 + 2*x)]];
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] = (((n^2 + 3*n + 2)*x + (2*n^2 + 6*n + 4)*x^2 + (n^2 + 3*n + 2)*x^3)*v[i - 2] + (2 + 6*x + 4*x^2)*v[i - 1])/(2 + 2*x)); 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) = T(n-1, k) + T(n-1, k-1) + ((n+1)*(n+2)/2)*T(n-2, k-1), read by rows.?
  - 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) = T(n-1, k) + T(n-1, k-1) + ((n+1)*(n+2)/2)*T(n-2, k-1), read by rows.?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Liu–Wang theorem with a lag row (above)

Related ?

Same fingerprint ?

A154227 → 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