A147644 → basis A008277

nonnegative coefficients

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

Recurrence

\(\left(-3 + n\right)\,P_{n}(x) = \left(-6 - 3\,x + 2\,n + n x\right)\,P_{n-1}(x) + \left(-3\,x + n x\right)\,P_{n-1}'(x)\)

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

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

\(P_{2}(x) = 4 + 5\,x + x^{2}\)

\(P_{3}(x) = 12 + 27\,x + 11\,x^{2} + x^{3}\)

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	4	5	1													10
3	12	27	11	1												51
4	24	93	71	16	1											205
5	48	303	377	151	22	1										902
6	96	957	1811	1132	283	29	1									4309
7	192	2967	8201	7471	2830	486	37	1								22185
8	384	9093	35771	45556	24451	6232	782	46	1							122316
9	768	27663	152177	263551	192262	68075	12488	1196	56	1						718237
10	1536	83757	636371	1469932	1417123	668787	167979	23252	1756	67	1					4470561
11	3072	252807	2629241	7986031	9972670	6098632	2012619	377247	40812	2493	79	1				29375704
12	6144	761493	10769771	42559396	67822051	52663094	22199584	5407842	785367	68235	3441	92	1			203046511
13	12288	2290623	43840577	223566751	449491702	436463709	230259766	70870162	13261512	1535952	109527	4637	106	1		1471707313
14	24576	6884157	177652931	1161674332	2920516963	3504737665	2278541837	868091224	203485282	30156984	2850276	169808	6121	121	1	11154792278

Technical details

Node id: n-9830a1a3d1e12981 · revision 0 · JSON
Definition source: derived:basis_map_A008277
Fingerprint: f6-b80b2223b98a405f

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,     5,      1;
  3:  12,    27,     11,      1;
  4:  24,    93,     71,     16,      1;
  5:  48,   303,    377,    151,     22,     1;
  6:  96,   957,   1811,   1132,    283,    29,     1;
  7: 192,  2967,   8201,   7471,   2830,   486,    37,    1;
  8: 384,  9093,  35771,  45556,  24451,  6232,   782,   46,  1;
  9: 768, 27663, 152177, 263551, 192262, 68075, 12488, 1196, 56, 1;

%F Formula

The row polynomials satisfy (n-3)*P(n,x) = (2*n-6+(n-3)*x)*P(n-1,x) + (n-3)*x*d/dx P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + x, P(2,x) = 4 + 5*x + x^2, P(3,x) = 12 + 27*x + 11*x^2 + x^3.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + x;
P[2, x_] := 4 + 5*x + x^2;
P[3, x_] := 12 + 27*x + 11*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 6 + (n - 3)*x)*P[n - 1, x] + (n - 3)*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] = 4 + 5*x + x^2; v[4] = 12 + 27*x + 11*x^2 + x^3; for(n = 4, N, my(i = n + 1); v[i] = ((2*n - 6 + (n - 3)*x)*v[i - 1] + (n - 3)*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) if n < 3, otherwise T(n, k) = binomial(n, k) + 2*binomial(n-2, k-1).?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.