A038330 → basis A013610 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(28 + 36\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 784 + 2016\,x + 1296\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	784	2016	1296								4096
1	21952	84672	108864	46656							262144
2	614656	3161088	6096384	5225472	1679616						16777216
3	17210368	110638080	284497920	365783040	235146240	60466176					1073741824
4	481890304	3717439488	11948912640	20483850240	19752284160	10158317568	2176782336				68719476736
5	13492928512	121436356608	468397375488	1003708661760	1290482565120	995515121664	426649337856	78364164096			4398046511104
6	377801998336	3885963411456	17486835351552	44966148046848	72267023646720	74331795750912	47784725839872	17553572757504	2821109907456		281474976710656
7	10578455953408	122407847460864	629526072655872	1888578217967616	3642257991794688	4682903132307456	4013916970549248	2211750167445504	710919696678912	101559956668416	18014398509481984

All 15 rows

Technical details

Node id: n-b0181b55e9572fd7 · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-27210632b0078bca

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:              784,               2016,               1296;
  1:            21952,              84672,             108864,               46656;
  2:           614656,            3161088,            6096384,             5225472,             1679616;
  3:         17210368,          110638080,          284497920,           365783040,           235146240,             60466176;
  4:        481890304,         3717439488,        11948912640,         20483850240,         19752284160,          10158317568,           2176782336;
  5:      13492928512,       121436356608,       468397375488,       1003708661760,       1290482565120,         995515121664,         426649337856,          78364164096;
  6:     377801998336,      3885963411456,     17486835351552,      44966148046848,      72267023646720,       74331795750912,       47784725839872,       17553572757504,        2821109907456;
  7:   10578455953408,    122407847460864,    629526072655872,    1888578217967616,    3642257991794688,     4682903132307456,     4013916970549248,     2211750167445504,      710919696678912,     101559956668416;
  8:  296196766695424,   3808244143226880,  22033412542955520,   75543128718704640,  169972039617085440,   262242575409217536,   280974187938447360,   206430015628247040,    99528757535047680,   28436787867156480,    3656158440062976;
  9: 8293509467471872, 117293919611387904, 754032340358922240, 2908410455670128640, 7478769743151759360, 13461785537673166848, 17308009977008357376, 15895111203375022080, 10218285773598228480, 4379265331542097920, 1126096799539396608, 131621703842267136;

%F Formula

The row polynomials satisfy P(n,x) = (28+36*x)*P(n-1,x) for n >= 1, with P(0,x) = 784 + 2016*x + 1296*x^2.

%t Mathematica

P[0, x_] := 784 + 2016*x + 1296*x^2;
P[n_, x_] := P[n, x] = Expand[(28 + 36*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 784 + 2016*x + 1296*x^2; for(n = 1, N, my(i = n + 1); v[i] = (28 + 36*x)*v[i - 1]); 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 Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

Related ?

Same fingerprint ?

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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.