A038281 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(11 + 3\,x\right)\,P_{n-1}(x) + \left(3\,x\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 121 + 75\,x + 9\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	121	75	9								205
1	1331	1413	378	27							3149
2	14641	23775	10665	1674	81						50836
3	161051	376773	252630	65475	6885	243					863057
4	1771561	5757975	5425029	2067390	354780	26973	729				15404437
5	19487171	85926333	109499418	57622887	14362110	1765638	102060	2187			288767804
6	214358881	1261430175	2119269105	1480955994	503197191	88992918	8256654	376164	6561		5676843643
7	2357947691	18303099093	39811865310	35976927195	16016403375	3823407441	506421720	36807210	1358127	19683	116834256845

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-db2ea4f508716e1f · revision 1 · JSON
Definition source: derived:basis_map_A008277
Fingerprint: f6-517bb25c95c6b465

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 2)
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:          121,            75,              9;
  1:         1331,          1413,            378,             27;
  2:        14641,         23775,          10665,           1674,             81;
  3:       161051,        376773,         252630,          65475,           6885,           243;
  4:      1771561,       5757975,        5425029,        2067390,         354780,         26973,           729;
  5:     19487171,      85926333,      109499418,       57622887,       14362110,       1765638,        102060,         2187;
  6:    214358881,    1261430175,     2119269105,     1480955994,      503197191,      88992918,       8256654,       376164,        6561;
  7:   2357947691,   18303099093,    39811865310,    35976927195,    16016403375,    3823407441,     506421720,     36807210,     1358127,     19683;
  8:  25937424601,  263317230375,   731711007549,   838974139830,   476308059210,  147457803591,   26156452203,   2697095880,   157956075,   4822335,    59049;
  9: 285311670611, 3764253499053, 13229038819458, 18974615819247, 13472007781320, 5262827070996, 1200910524660, 164776424769, 13619750265, 657116955, 16888014, 177147;

%F Formula

The row polynomials satisfy P(n,x) = (11+3*x)*P(n-1,x) + 3*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 121 + 75*x + 9*x^2.

%t Mathematica

P[0, x_] := 121 + 75*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[(11 + 3*x)*P[n - 1, x] + 3*x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 121 + 75*x + 9*x^2; for(n = 1, N, my(i = n + 1); v[i] = (11 + 3*x)*v[i - 1] + 3*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 - 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-d7f363249f6694ec
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-d7f363249f6694ec
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-d7f363249f6694ec

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-d7f363249f6694ec

Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ?
- premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*3^j.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-da334b7c58401d78 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - parent_provers periodic_product e-c5656743efdffc45 root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

No candidate equalities or related sequences.

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