A146986 → basis A008277

nonnegative coefficients

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

Recurrence

\(\left(-3 + n\right)\,P_{n}(x) = \left(24 + 30\,x + 6\,x^{2} - 8\,n - 10\,n x - 2\,n x^{2}\right)\,P_{n-2}(x) + \left(30\,x + 12\,x^{2} - 10\,n x - 4\,n x^{2}\right)\,P_{n-2}'(x) + \left(6\,x^{2} - 2\,n x^{2}\right)\,P_{n-2}''(x) + \left(-18 - 9\,x + 6\,n + 3\,n x\right)\,P_{n-1}(x) + \left(-9\,x + 3\,n x\right)\,P_{n-1}'(x)\)

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

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

\(P_{2}(x) = 6 + 7\,x + x^{2}\)

\(P_{3}(x) = 16 + 35\,x + 13\,x^{2} + x^{3}\)

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	6	7	1													14
3	16	35	13	1												65
4	48	177	119	22	1											367
5	160	947	1021	333	36	1										2498
6	576	5337	8711	4462	853	59	1									19999
7	2176	31115	74293	56141	16780	2098	99	1								182703
8	8448	184737	630239	674422	293461	57514	5066	172	1							1854060
9	33280	1106147	5299741	7800493	4713716	1334011	184954	12118	310	1						20484771
10	132096	6645417	44144471	87471742	71021413	27540177	5480139	566912	28786	577	1					243031731
11	526336	39961595	364525573	956877261	1019202780	522517276	141021993	20827467	1671924	67901	1101	1				3067201208
12	2101248	240294897	2988268559	10265257222	14090072821	9310033568	3282373732	652162764	74379987	4774035	158937	2138	1			40909879909
13	8396800	1444385747	24354774061	108458417453	189254131396	158161203541	70942331328	18241683856	2780015370	252375332	13259455	369003	4200	1		573911347543
14	33570816	8678341497	197597521031	1132408299022	2485652178373	2589891971095	1449474572117	469481316304	91902582772	11084830614	820326936	35951918	849691	8311	1	8437062320498

Technical details

Node id: n-fc3066c48ace6df3 · revision 0 · JSON
Definition source: derived:basis_map_A008277
Fingerprint: f6-a7266729c32b025a

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:     6,       7,       1;
  3:    16,      35,      13,       1;
  4:    48,     177,     119,      22,       1;
  5:   160,     947,    1021,     333,      36,       1;
  6:   576,    5337,    8711,    4462,     853,      59,      1;
  7:  2176,   31115,   74293,   56141,   16780,    2098,     99,     1;
  8:  8448,  184737,  630239,  674422,  293461,   57514,   5066,   172,   1;
  9: 33280, 1106147, 5299741, 7800493, 4713716, 1334011, 184954, 12118, 310, 1;

%F Formula

The row polynomials satisfy (n-3)*P(n,x) = -(8*n-24+(10*n-30)*x+(2*n-6)*x^2)*P(n-2,x) - ((10*n-30)*x+(4*n-12)*x^2)*d/dx P(n-2,x) - (2*n-6)*x^2*d^2/dx^2 P(n-2,x) + (6*n-18+(3*n-9)*x)*P(n-1,x) + (3*n-9)*x*d/dx P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + x, P(2,x) = 6 + 7*x + x^2, P(3,x) = 16 + 35*x + 13*x^2 + x^3.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + x;
P[2, x_] := 6 + 7*x + x^2;
P[3, x_] := 16 + 35*x + 13*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(8*n - 24 + (10*n - 30)*x + (2*n - 6)*x^2)*P[n - 2, x] - ((10*n - 30)*x + (4*n - 12)*x^2)*D[P[n - 2, x], x] - (2*n - 6)*x^2*D[P[n - 2, x], {x, 2}] + (6*n - 18 + (3*n - 9)*x)*P[n - 1, x] + (3*n - 9)*x*D[P[n - 1, x], x])/(n - 3)]];
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 + x; v[3] = 6 + 7*x + x^2; v[4] = 16 + 35*x + 13*x^2 + x^3; for(n = 4, N, my(i = n + 1); v[i] = (-(8*n - 24 + (10*n - 30)*x + (2*n - 6)*x^2)*v[i - 2] - ((10*n - 30)*x + (4*n - 12)*x^2)*deriv(v[i - 2], x) - (2*n - 6)*x^2*deriv(deriv(v[i - 2], x), x) + (6*n - 18 + (3*n - 9)*x)*v[i - 1] + (3*n - 9)*x*deriv(v[i - 1], x))/(n - 3)); 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, read by rows, T(n, k) = binomial(n, k) for n < 2 and binomial(n, k) + 2^(n-1) * binomial(n-2, k-1) otherwise.?
  - 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