A257616 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(2 + 8\,x + 10\,x^{2} + 4\,x^{3}\right)\,P_{n-1}(x) + \left(6\,x + 18\,x^{2} + 18\,x^{3} + 6\,x^{4}\right)\,P_{n-1}'(x)\)

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

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	40	40													84
3	8	336	960	640												1944
4	16	2720	16800	28160	14080											61776
5	32	21824	262400	832000	985600	394240										2496096
6	64	174720	3891840	20838400	43929600	40212480	13404160									122451264
7	128	1398016	56232960	479037440	1600614400	2516828160	1876582400	536166400								7066859904
8	256	11184640	801241600	10480476160	52154798080	125355704320	156882288640	98654617600	24663654400							469003965696
9	512	89478144	11329228800	222429388800	1586595225600	5471716884480	10223620915200	10616094720000	5771295129600	1282510028800						35185680999936
10	1024	715827200	159503984640	4629855436800	46144922419200	219519606620160	574535968849920	876053004288000	776905113600000	371927908352000	74385581670400					2944262181049344
11	2048	5726621696	2240214056960	95149172490240	1301624802508800	8316685239582720	29296033441382400	61527770942668800	79143693877248000	61227028774912000	26183724747980800	4760677226905600				271854634166360064
12	4096	45812981760	31420263014400	1938826874716160	35935526660014080	302579422136893440	1396016568918343680	3879063259132723200	6787462157225164800	7544185693011968000	5174558603319705600	1999484435300352000	333247405883392000			27454503364539273216
13	8192	366503870464	440341812019200	39279261702553600	976977884404121600	10688736254860984320	63336329971551436800	226519446158573568000	517810017781363507200	775422430984378777600	758385403600515891200	467136692212885094400	164624218506395648000	25326802847137792000		3010266776172085272576
14	16384	2932030996480	6168450406973440	792630703043379200	26265568751963340800	369394940918866903040	2770197571584228065280	12500308829839294464000	36310395412362559488000	70349777059756113920000	91994396020325954355200	80301960900909203456000	44881760624375234560000	14537584834257092608000	2076797833465298944000	356119638398631291469824

Technical details

Node id: n-78dc37aeb5f3a608 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-72ea6b42576fa2d7

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,       40,          40;
  3:   8,      336,         960,          640;
  4:  16,     2720,       16800,        28160,         14080;
  5:  32,    21824,      262400,       832000,        985600,        394240;
  6:  64,   174720,     3891840,     20838400,      43929600,      40212480,       13404160;
  7: 128,  1398016,    56232960,    479037440,    1600614400,    2516828160,     1876582400,      536166400;
  8: 256, 11184640,   801241600,  10480476160,   52154798080,  125355704320,   156882288640,    98654617600,   24663654400;
  9: 512, 89478144, 11329228800, 222429388800, 1586595225600, 5471716884480, 10223620915200, 10616094720000, 5771295129600, 1282510028800;

%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) + (6*x+18*x^2+18*x^3+6*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] + (6*x + 18*x^2 + 18*x^3 + 6*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] + (6*x + 18*x^2 + 18*x^3 + 6*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) = 6*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) = 6*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