A133080 → basis A008277

nonnegative coefficients

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence

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

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

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

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	1														3
2	1	3	1													5
3	2	10	7	1												20
4	1	15	25	10	1											52
5	2	46	115	75	16	1										255
6	1	63	301	350	140	21	1									877
7	2	190	1267	2051	1190	287	29	1								5017
8	1	255	3025	7770	6951	2646	462	36	1							21147
9	2	766	12355	41875	49476	25473	6342	786	46	1						137122
10	1	1023	28501	145750	246730	179487	63987	11880	1155	55	1					678570
11	2	3070	115027	757251	1626130	1503139	691383	170907	23430	1760	67	1				4892167
12	1	4095	261625	2532530	7508501	9321312	5715424	1899612	359502	39325	2431	78	1			27644437
13	2	12286	1050595	12924275	47583536	72757685	55044704	22811932	5494632	792077	68497	3445	92	1		218543759
14	1	16383	2375101	42355950	210766920	420693273	408741333	216627840	67128490	12662650	1479478	106470	4550	105	1	1382958545

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Technical details

Node id: n-5a62d17a58e2a9f3 · revision 1 · JSON
Definition source: derived:basis_map_A008277
Fingerprint: f6-07f01b90c025cde2

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

OEIS matches ?

No OEIS matches.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0: 1;
  1: 2,   1;
  2: 1,   3,     1;
  3: 2,  10,     7,     1;
  4: 1,  15,    25,    10,     1;
  5: 2,  46,   115,    75,    16,     1;
  6: 1,  63,   301,   350,   140,    21,    1;
  7: 2, 190,  1267,  2051,  1190,   287,   29,   1;
  8: 1, 255,  3025,  7770,  6951,  2646,  462,  36,  1;
  9: 2, 766, 12355, 41875, 49476, 25473, 6342, 786, 46, 1;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[(1 + 3*x + x^2)*P[n - 2, x] + (3*x + 2*x^2)*D[P[n - 2, x], x] + x^2*D[P[n - 2, x], {x, 2}]];
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; for(n = 2, N, my(i = n + 1); v[i] = (1 + 3*x + x^2)*v[i - 2] + (3*x + 2*x^2)*deriv(v[i - 2], x) + x^2*deriv(deriv(v[i - 2], x), x)); 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)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-5d454ac027f458ad
Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
All coefficients of each polynomial have the same sign (after x -> -x when alternating).

derived from the cited steps by a stated theorem: e-5d454ac027f458ad
Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.

derived from the cited steps by a stated theorem: e-5d454ac027f458ad

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

basis_map A008277 e-5d454ac027f458ad

Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ?
- premise (sequence): Interpolation operator: Triangle with an even number of zeros in each line followed by 1 or 2 ones.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers periodic_product e-4d02ac33f5b2fbc9 root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

No candidate equalities or related sequences.

References

[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