A038305 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(13 + 3\,x\right)\,P_{n-1}(x) + \left(3\,x\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 169 + 87\,x + 9\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	169	87	9								265
1	2197	1899	432	27							4555
2	28561	36975	13905	1890	81						81412
3	371293	677283	375120	83295	7695	243					1514929
4	4826809	11950407	9159129	2957850	442260	29889	729				29367073
5	62748517	205686939	209874672	92550087	19930050	2163672	112266	2187			593068390
6	815730721	3479236575	4604679585	2665725930	775901511	120372966	9971262	411156	6561		12472036267
7	10604499373	58114977363	97926621840	72460009215	27394715565	5698147581	670228020	43893090	1476225	19683	272914587955

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-8540506b111a5858 · revision 1 · JSON
Definition source: derived:basis_map_A008277
Fingerprint: f6-86d4933b561cba07

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:           169,             87,              9;
  1:          2197,           1899,            432,             27;
  2:         28561,          36975,          13905,           1890,             81;
  3:        371293,         677283,         375120,          83295,           7695,           243;
  4:       4826809,       11950407,        9159129,        2957850,         442260,         29889,           729;
  5:      62748517,      205686939,      209874672,       92550087,       19930050,       2163672,        112266,         2187;
  6:     815730721,     3479236575,     4604679585,     2665725930,      775901511,     120372966,       9971262,       411156,        6561;
  7:   10604499373,    58114977363,    97926621840,    72460009215,    27394715565,    5698147581,     670228020,     43893090,     1476225,     19683;
  8:  137858491849,   961653135927,  2034950747049,  1887900068250,   902247916770,  241732278963,   37871511363,   3503049120,   186299595,   5215995,    59049;
  9: 1792160394037, 15800025650379, 41549023601712, 47638653742647, 28219898124000, 9475247561274, 1899213689142, 232718204169, 17402232375, 767538585, 18187092, 177147;

%F Formula

The row polynomials satisfy P(n,x) = (13+3*x)*P(n-1,x) + 3*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 169 + 87*x + 9*x^2.

%t Mathematica

P[0, x_] := 169 + 87*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[(13 + 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] = 169 + 87*x + 9*x^2; for(n = 1, N, my(i = n + 1); v[i] = (13 + 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-1eab37781c60e76a
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-1eab37781c60e76a
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-1eab37781c60e76a

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-1eab37781c60e76a

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)*10^(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-e186153c671b61f0 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - parent_provers periodic_product e-f733aee5869de636 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