A008556 → basis A008277

nonnegative coefficients

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

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2								2
2	8	2							10
3	32	12							44
4	136	78	6						220
5	592	460	60						1112
6	2624	2660	540	20					5844
7	11776	15064	4200	280					31320
8	53344	84294	30730	3220	70				171658
9	243392	467436	214452	31080	1260				957620
10	1116928	2575212	1450260	275100	17640	252			5435392
11	5149696	14117224	9576600	2282280	203280	5544			31334624
12	23835904	77096316	62105604	18101160	2111340	91476	924		183342724
13	110690816	419788824	397072104	138687120	20300280	1225224	24024		1087788392
14	515483648	2280434312	2509955448	1034545512	184744560	14606592	456456	3432	6540229960

Technical details

Node id: n-f8e557445427987b · revision 0 · JSON
Definition source: derived:basis_map_A008277
Fingerprint: f6-5f44c6b5e115ca0a

Observed on the stored rows ?

Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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:      8,      2;
  3:     32,     12;
  4:    136,     78,      6;
  5:    592,    460,     60;
  6:   2624,   2660,    540,    20;
  7:  11776,  15064,   4200,   280;
  8:  53344,  84294,  30730,  3220,   70;
  9: 243392, 467436, 214452, 31080, 1260;

%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)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2;
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)*P[n - 1, 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; 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)*v[i - 1])/(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 Triangle of coefficients of Legendre polynomials 2^n P_n (x).?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? 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