Hadamard-factor parent of A155495 → basis A008277

nonnegative coefficients

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

Recurrence

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

\(P_{1}(x) = 8 + 9\,x + x^{2}\)

\(P_{2}(x) = 32 + 67\,x + 21\,x^{2} + x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
1	8	9	1															18
2	32	67	21	1														121
3	128	449	263	38	1													879
4	512	2851	2685	725	60	1												6834
5	2048	17553	24655	10514	1625	87	1											56483
6	8192	106051	212541	131649	31808	3178	119	1										493539
7	32768	633089	1759103	1506118	520741	80794	5642	156	1									4538412
8	131072	3749827	14162493	16225789	7618116	1675227	180810	9318	198	1								43752851
9	524288	22095249	111831151	167599210	103235185	30476773	4630827	367560	14550	245	1							440775039
10	2097152	129739171	870809181	1679060009	1324387768	506475684	102492005	11395395	692868	21725	297	1						4627171256
11	8388608	760009793	6711817151	16442574558	16313984653	7880799912	2047633988	301413156	25572723	1228931	31273	354	1					50493455101
12	33554432	4444999651	51340005405	158254054725	194838531420	116698726053	37938411160	7123458680	796514290	53248052	2073071	43667	416	1				571523621023
13	134217728	25968708369	390474355951	1502918291490	2271736892985	1663170793347	663694423413	154829792160	21971717560	1928792866	104200096	3352986	59423	483	1			6696935598858
14	536870912	151602131587	2957003689341	14124700107601	25990144407600	22998595903998	11101145041539	3155377559253	553326729600	61399584350	4342853658	193511578	5232500	79100	555	1		81098373703173
15	2147483648	884583605249	22319296011263	131656403343158	292872440422261	310432811309142	179155897609950	61121228036076	12984958480853	1772863102074	157988864878	9194302156	343657470	7917812	103300	632	1	1013370164249923

Technical details

Node id: n-e6dc6f414384bbe9 · revision 0 · JSON
Definition source: derived:basis_map_A008277
Fingerprint: f6-91622946028ad904

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 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:       8,         9,         1;
   2:      32,        67,        21,          1;
   3:     128,       449,       263,         38,          1;
   4:     512,      2851,      2685,        725,         60,         1;
   5:    2048,     17553,     24655,      10514,       1625,        87,         1;
   6:    8192,    106051,    212541,     131649,      31808,      3178,       119,        1;
   7:   32768,    633089,   1759103,    1506118,     520741,     80794,      5642,      156,      1;
   8:  131072,   3749827,  14162493,   16225789,    7618116,   1675227,    180810,     9318,    198,     1;
   9:  524288,  22095249, 111831151,  167599210,  103235185,  30476773,   4630827,   367560,  14550,   245,   1;
  10: 2097152, 129739171, 870809181, 1679060009, 1324387768, 506475684, 102492005, 11395395, 692868, 21725, 297, 1;

%F Formula

The row polynomials satisfy P(n,x) = -(x+x^2)*P(n-2,x) - (x+2*x^2)*d/dx P(n-2,x) - x^2*d^2/dx^2 P(n-2,x) + (4+2*x)*P(n-1,x) + 2*x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 8 + 9*x + x^2, P(2,x) = 32 + 67*x + 21*x^2 + x^3.

%t Mathematica

P[1, x_] := 8 + 9*x + x^2;
P[2, x_] := 32 + 67*x + 21*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[-(x + x^2)*P[n - 2, x] - (x + 2*x^2)*D[P[n - 2, x], x] - x^2*D[P[n - 2, x], {x, 2}] + (4 + 2*x)*P[n - 1, x] + 2*x*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] = 8 + 9*x + x^2; v[2] = 32 + 67*x + 21*x^2 + x^3; for(n = 3, N, my(i = n); v[i] = -(x + x^2)*v[i - 2] - (x + 2*x^2)*deriv(v[i - 2], x) - x^2*deriv(deriv(v[i - 2], x), x) + (4 + 2*x)*v[i - 1] + 2*x*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 Ray basis map theorem
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Ray basis map theorem
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 hadamard_factor parent of Triangle read by rows: t(n,m) = binomial(2*n,2*m) * binomial(n,m).?
  - Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
  - 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 hadamard_factor parent of Triangle read by rows: t(n,m) = binomial(2*n,2*m) * binomial(n,m).?
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)

Related ?

Same fingerprint ?

References

[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[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
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023