A115951 → basis A008277

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(-4 - 4\,x + 4\,n + 4\,n x\right)\,P_{n-2}(x) + \left(-4\,x + 4\,n x\right)\,P_{n-2}'(x) + \left(-2 - 2\,x + 4\,n + 4\,n x\right)\,P_{n-1}(x) + \left(-2\,x + 4\,n x\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 2 + 2\,x\)

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	2														4
2	8	20	6													34
3	32	176	132	20												360
4	136	1488	2116	760	70											4570
5	592	12432	30040	19240	4060	252										66616
6	2624	103712	402144	409520	148680	20664	924									1088268
7	11776	867456	5217856	7944832	4417840	1031856	101640	3432								19596688
8	53344	7285248	66539344	145880224	116662840	40737984	6632472	487344	12870							384291670
9	243392	61456064	840376032	2587328672	2861522160	1401841392	335489616	40243632	2290860	48620						8130840440
10	1116928	520636800	10558671616	44861041600	66825869920	44135694624	14564816112	2536179360	233444640	10599160	184756					184248255516
11	5149696	4427874816	132323885056	766155165952	1508490364480	1307253127488	571433259936	135470553504	17926914720	1306127680	48406072	705432				4444841534832
12	23835904	37788879360	1656752587648	12951631176704	33239291556416	37065381800832	20880257290816	6472723390848	1154785343712	120032667040	7094260888	218683920	2704156			113585984178244
13	110690816	323492530688	20743808427264	217432092543744	719709044875392	1017728888944768	724316066714240	285580900761728	65964352918464	9172541069120	768594567312	37588708976	978904472	10400600		3061778472057584
14	515483648	2776746681344	259884570519552	3633256290273792	15384176838541056	27278663623154944	24162617623464320	11875888788667904	3454076094049152	616979504261120	68725857616416	4740496455968	195012914096	4347450800	40116600	86741986349650712

Technical details

Node id: n-445f9a43ace6155b · revision 0 · JSON
Definition source: derived:basis_map_A008277
Fingerprint: f6-44562cdf8dd879d1

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,        2;
  2:      8,       20,         6;
  3:     32,      176,       132,         20;
  4:    136,     1488,      2116,        760,         70;
  5:    592,    12432,     30040,      19240,       4060,        252;
  6:   2624,   103712,    402144,     409520,     148680,      20664,       924;
  7:  11776,   867456,   5217856,    7944832,    4417840,    1031856,    101640,     3432;
  8:  53344,  7285248,  66539344,  145880224,  116662840,   40737984,   6632472,   487344,   12870;
  9: 243392, 61456064, 840376032, 2587328672, 2861522160, 1401841392, 335489616, 40243632, 2290860, 48620;

%F Formula

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

%t Mathematica

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

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((4*n - 4 + (4*n - 4)*x)*v[i - 2] + (4*n - 4)*x*deriv(v[i - 2], x) + (4*n - 2 + (4*n - 2)*x)*v[i - 1] + (4*n - 2)*x*deriv(v[i - 1], x))/(n)); 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 Ray basis map theorem
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Ray basis map theorem
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Ray basis map theorem

Proof chains ? JSON

Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›
- from Expansion of 1/sqrt(1-4*x*y-4*x^2*y).?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

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
[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741