A142075 → reverse

nonnegative coefficients

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	2	1														3
3	4	8	1													13
4	8	44	22	1												75
5	16	208	264	52	1											541
6	32	912	2416	1208	114	1										4683
7	64	3840	19056	19328	4764	240	1									47293
8	128	15808	137376	249904	124952	17172	494	1								545835
9	256	64256	934912	2823488	2499040	705872	58432	1004	1							7087261
10	512	259328	6123520	29132288	41931328	20965664	3641536	191360	2026	1						102247563
11	1024	1042432	39075072	282046464	623239296	503175936	155809824	17627904	610548	4072	1					1622632573
12	2048	4180992	244874752	2608047360	8488764672	10400786304	5200393152	1061095584	81501480	1913084	8166	1				28091567595
13	4096	16748544	1515239424	23313126400	108360072960	192719553024	145611008256	48179888256	6772504560	364267600	5918904	16356	1			526858348381
14	8192	67047424	9292419072	203172644864	1316731993600	3287875292928	3580310539776	1790155269888	410984411616	41147874800	1587286288	18149256	32738	1		10641342970443
15	16384	268304384	56626130944	1736890187776	15402219029504	52621823451136	79715225193216	57284968636416	19928806298304	3288863965696	240659672336	6784727296	55298956	65504	1	230283190977853

Technical details

Node id: n-ce7e2a3941a4e8f5 · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-735f72d4be30f803

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 -1)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:   1;
   2:   2,      1;
   3:   4,      8,       1;
   4:   8,     44,      22,        1;
   5:  16,    208,     264,       52,        1;
   6:  32,    912,    2416,     1208,      114,        1;
   7:  64,   3840,   19056,    19328,     4764,      240,       1;
   8: 128,  15808,  137376,   249904,   124952,    17172,     494,      1;
   9: 256,  64256,  934912,  2823488,  2499040,   705872,   58432,   1004,    1;
  10: 512, 259328, 6123520, 29132288, 41931328, 20965664, 3641536, 191360, 2026, 1;

%F Formula

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

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(2 + (n - 1)*x)*P[n - 1, x] + (2*x - x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (2 + (n - 1)*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, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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) = 2^(k-1) * E(n, k-1) where E(n,k) are the Eulerian numbers A173018, read by rows.?
  - 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