A257618 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(2 + 8\,x + 10\,x^{2} + 4\,x^{3}\right)\,P_{n-1}(x) + \left(8\,x + 24\,x^{2} + 24\,x^{3} + 8\,x^{4}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	4														6
2	4	48	48													100
3	8	496	1440	960												2904
4	16	4992	31872	53760	26880											117520
5	32	49984	633600	2035200	2419200	967680										6105696
6	64	499968	12004608	65587200	139238400	127733760	42577920									387641920
7	128	4999936	222082560	1945359360	6570547200	10377400320	7749181440	2214051840								29083622784
8	256	49999872	4057485312	55020994560	277868666880	672390512640	844064686080	531372441600	132843110400							2517667897600
9	512	499999744	73634734080	1511695564800	10988122521600	38243673538560	71788416860160	74710965288960	40649991782400	9033331507200						247000331798016
10	1024	4999999488	1331425210368	40776779366400	415923641548800	2001806699397120	5272142478704640	8066233663488000	7165557374976000	3432665972736000	686533194547200					27082976229975040
11	2048	49999998976	24025653780480	1086824767733760	15283153634918400	99049132470435840	351686618708705280	741992961374945280	956900806557696000	741275183480832000	317178335880806400	57668788341964800				3282145880871819264
12	4096	499999997952	433061768036352	28737957036687360	550058317083770880	4710257094615367680	21942492764134440960	61323295932599500800	107675030915304652800	119923618423504896000	82345537486769356800	31833171164764569600	5305528527460761600			435638162145042042880
13	8192	4999999995904	7801111824629760	755848118314598400	19506645577875456000	217632897388861194240	1304375950369798225920	4697760787825761976320	10785949796366077132800	16196249865580078694400	15866529074101459353600	9782010553717437235200	3448593542849495040000	530552852746076160000		62849925620753059700736
14	16384	49999999991808	140480012843286528	19808073312672153600	684389696960574259200	9842820931135686574080	74794645003599803842560	340297675113123702374400	993738333829706809344000	1931967076205819068416000	2531970373860405647769600	2213169392059091936870400	1237907138141254739558400	401097956676033576960000	57299708096576225280000	9792789458217033286696960

Technical details

Node id: n-2e3a2796ffe64916 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-d6ac9abcd3bcb1be

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:   2,         4;
  2:   4,        48,          48;
  3:   8,       496,        1440,           960;
  4:  16,      4992,       31872,         53760,          26880;
  5:  32,     49984,      633600,       2035200,        2419200,         967680;
  6:  64,    499968,    12004608,      65587200,      139238400,      127733760,       42577920;
  7: 128,   4999936,   222082560,    1945359360,     6570547200,    10377400320,     7749181440,     2214051840;
  8: 256,  49999872,  4057485312,   55020994560,   277868666880,   672390512640,   844064686080,   531372441600,   132843110400;
  9: 512, 499999744, 73634734080, 1511695564800, 10988122521600, 38243673538560, 71788416860160, 74710965288960, 40649991782400, 9033331507200;

%F Formula

The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (2+8*x+10*x^2+4*x^3)*P(n-1,x) + (8*x+24*x^2+24*x^3+8*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2 + 8*x + 10*x^2 + 4*x^3)*P[n - 1, x] + (8*x + 24*x^2 + 24*x^3 + 8*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; for(n = 1, N, my(i = n + 1); v[i] = ((2 + 8*x + 10*x^2 + 4*x^3)*v[i - 1] + (8*x + 24*x^2 + 24*x^3 + 8*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 read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 8*x + 2.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 8*x + 2.?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

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