A144696 → reverse

nonnegative coefficients

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

Recurrence

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

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

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	1														3
2	4	7	1													12
3	8	33	18	1												60
4	16	131	171	41	1											360
5	32	473	1208	718	88	1										2520
6	64	1611	7197	8422	2682	183	1									20160
7	128	5281	38454	78095	49780	9327	374	1								181440
8	256	16867	190783	621199	689155	264409	30973	757	1							1814400
9	512	52905	898068	4441476	7862124	5296638	1305420	99732	1524	1						19958400
10	1024	163835	4068417	29391924	78263076	84249210	36926550	6119268	314436	3059	1					239500800
11	2048	502769	17912018	183575373	704713848	1137586002	800907660	239706162	27621432	976957	6130	1				3113510400
12	4096	1532883	77178531	1096997045	5880461445	13600812798	14370363294	6962801418	1474745130	121232255	3004431	12273	1			43589145600
13	8192	4651897	327108720	6333949066	46252739120	148129842591	223769408736	163257755820	56524259808	8707280455	520982192	9172842	24560	1		653837184000
14	16384	14070379	1368909541	35595049970	347189874446	1499436289337	3123323853207	3259475072268	1708046888820	434925643853	49788188579	2203175714	27862366	49135	1	10461394944000

Technical details

Node id: n-5823146f9baf5bf7 · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-e57a087584e4898e

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,     1;
  2:   4,     7,      1;
  3:   8,    33,     18,       1;
  4:  16,   131,    171,      41,       1;
  5:  32,   473,   1208,     718,      88,       1;
  6:  64,  1611,   7197,    8422,    2682,     183,       1;
  7: 128,  5281,  38454,   78095,   49780,    9327,     374,     1;
  8: 256, 16867, 190783,  621199,  689155,  264409,   30973,   757,    1;
  9: 512, 52905, 898068, 4441476, 7862124, 5296638, 1305420, 99732, 1524, 1;

%F Formula

The row polynomials satisfy P(n,x) = (2+n*x)*P(n-1,x) + (x-x^2)*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[(2 + n*x)*P[n - 1, x] + (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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (2 + n*x)*v[i - 1] + (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 Triangle of 2-Eulerian numbers.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check

Related ?

Same fingerprint ?

A144696: Triangle of 2-Eulerian numbers.

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