A038329 → basis A013610 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(24 + 27\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 576 + 1296\,x + 729\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	576	1296	729								2601
1	13824	46656	52488	19683							132651
2	331776	1492992	2519424	1889568	531441						6765201
3	7962624	44789760	100776960	113374080	63772920	14348907					345025251
4	191102976	1289945088	3627970560	5441955840	4591650240	2066242608	387420489				17596287801
5	4586471424	36118462464	121899810816	228562145280	257132413440	173564379072	65086642152	10460353203			897410677851
6	110075314176	990677827584	3900793946112	8776786378752	12342355845120	11108120260608	6248317646592	2008387814976	282429536481		45767944570401
7	2641807540224	26748301344768	120367356051456	315964309635072	533189772509184	599838494072832	449878870554624	216905884017408	61004779879896	7625597484987	2334165173090451

All 15 rows

Technical details

Node id: n-c67f40b153a1102f · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-cb7716efe3e0f2cd

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:              576,              1296,                729;
  1:            13824,             46656,              52488,              19683;
  2:           331776,           1492992,            2519424,            1889568,             531441;
  3:          7962624,          44789760,          100776960,          113374080,           63772920,            14348907;
  4:        191102976,        1289945088,         3627970560,         5441955840,         4591650240,          2066242608,           387420489;
  5:       4586471424,       36118462464,       121899810816,       228562145280,       257132413440,        173564379072,         65086642152,         10460353203;
  6:     110075314176,      990677827584,      3900793946112,      8776786378752,     12342355845120,      11108120260608,       6248317646592,       2008387814976,       282429536481;
  7:    2641807540224,    26748301344768,    120367356051456,    315964309635072,    533189772509184,     599838494072832,     449878870554624,     216905884017408,     61004779879896,      7625597484987;
  8:   63403380965376,   713288035860480,   3611020681543680,  10833062044631040,  21327590900367360,   28792247715495936,   26992732233277440,   17352470721392640,   7320573585587520,   1830143396396880,   205891132094649;
  9: 1521681143169024, 18830804146716672, 105923273325281280, 357491047472824320, 804354856813854720, 1266858899481821184, 1425216261917048832, 1145263067611914240, 644210475531701760, 241578928324388160, 54355258872987336, 5559060566555523;

%F Formula

The row polynomials satisfy P(n,x) = (24+27*x)*P(n-1,x) for n >= 1, with P(0,x) = 576 + 1296*x + 729*x^2.

%t Mathematica

P[0, x_] := 576 + 1296*x + 729*x^2;
P[n_, x_] := P[n, x] = Expand[(24 + 27*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] = 576 + 1296*x + 729*x^2; for(n = 1, N, my(i = n + 1); v[i] = (24 + 27*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)*3^j.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.?
    
    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
    
    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
    
    Parent-row provers of polytool 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.