A156365 → reverse

nonnegative coefficients

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

Recurrence

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

\(P_{0}(x) = 4 + 8\,x + x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	4	8	1															13
1	8	44	22	1														75
2	16	208	264	52	1													541
3	32	912	2416	1208	114	1												4683
4	64	3840	19056	19328	4764	240	1											47293
5	128	15808	137376	249904	124952	17172	494	1										545835
6	256	64256	934912	2823488	2499040	705872	58432	1004	1									7087261
7	512	259328	6123520	29132288	41931328	20965664	3641536	191360	2026	1								102247563
8	1024	1042432	39075072	282046464	623239296	503175936	155809824	17627904	610548	4072	1							1622632573
9	2048	4180992	244874752	2608047360	8488764672	10400786304	5200393152	1061095584	81501480	1913084	8166	1						28091567595
10	4096	16748544	1515239424	23313126400	108360072960	192719553024	145611008256	48179888256	6772504560	364267600	5918904	16356	1					526858348381
11	8192	67047424	9292419072	203172644864	1316731993600	3287875292928	3580310539776	1790155269888	410984411616	41147874800	1587286288	18149256	32738	1				10641342970443
12	16384	268304384	56626130944	1736890187776	15402219029504	52621823451136	79715225193216	57284968636416	19928806298304	3288863965696	240659672336	6784727296	55298956	65504	1			230283190977853
13	32768	1073463296	343513047040	14631261204480	174864872548352	800886290738176	1642231387216384	1633996524921600	816998262460800	205278923402048	25027696585568	1366131816784	28576682040	167730980	131038	1		5315654681981355
14	65536	4294377472	2077180231680	121859272294400	1938855121141760	11709013959438336	31800988619149312	42566258270909440	29411937448588800	10641564567727360	1987561788696832	182953343116224	7573652816960	119003195600	507124080	262108	1	130370767029135901

Technical details

Node id: n-a2b074a65f23d202 · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-735f72d4be30f803

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
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:    4,       8,         1;
  1:    8,      44,        22,          1;
  2:   16,     208,       264,         52,          1;
  3:   32,     912,      2416,       1208,        114,           1;
  4:   64,    3840,     19056,      19328,       4764,         240,          1;
  5:  128,   15808,    137376,     249904,     124952,       17172,        494,          1;
  6:  256,   64256,    934912,    2823488,    2499040,      705872,      58432,       1004,        1;
  7:  512,  259328,   6123520,   29132288,   41931328,    20965664,    3641536,     191360,     2026,       1;
  8: 1024, 1042432,  39075072,  282046464,  623239296,   503175936,  155809824,   17627904,   610548,    4072,    1;
  9: 2048, 4180992, 244874752, 2608047360, 8488764672, 10400786304, 5200393152, 1061095584, 81501480, 1913084, 8166, 1;

%F Formula

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

%t Mathematica

P[0, x_] := 4 + 8*x + x^2;
P[n_, x_] := P[n, x] = Expand[(2 + (n + 2)*x)*P[n - 1, x] + (2*x - x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 4 + 8*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = (2 + (n + 2)*x)*v[i - 1] + (2*x - x^2)*deriv(v[i - 1], 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 (−∞, 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 T(n, k) = E(n, k)*2^k where E(n,k) are the Eulerian numbers A173018, for n > 0 and 0 <= k <= n-1, additionally T(0,0) =…?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check

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