A038329 → basis A013611 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

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

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

Rows ?

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

All 15 rows

Technical details

Node id: n-e4f29e5c64f52699 · revision 0 · JSON
Definition source: derived:basis_map_A013609
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:              729,              1296,                576;
  1:            19683,             52488,              46656,              13824;
  2:           531441,           1889568,            2519424,            1492992,              331776;
  3:         14348907,          63772920,          113374080,          100776960,            44789760,             7962624;
  4:        387420489,        2066242608,         4591650240,         5441955840,          3627970560,          1289945088,           191102976;
  5:      10460353203,       65086642152,       173564379072,       257132413440,        228562145280,        121899810816,         36118462464,         4586471424;
  6:     282429536481,     2008387814976,      6248317646592,     11108120260608,      12342355845120,       8776786378752,       3900793946112,       990677827584,       110075314176;
  7:    7625597484987,    61004779879896,    216905884017408,    449878870554624,     599838494072832,     533189772509184,     315964309635072,    120367356051456,     26748301344768,      2641807540224;
  8:  205891132094649,  1830143396396880,   7320573585587520,  17352470721392640,   26992732233277440,   28792247715495936,   21327590900367360,  10833062044631040,   3611020681543680,    713288035860480,    63403380965376;
  9: 5559060566555523, 54355258872987336, 241578928324388160, 644210475531701760, 1145263067611914240, 1425216261917048832, 1266858899481821184, 804354856813854720, 357491047472824320, 105923273325281280, 18830804146716672, 1521681143169024;

%F Formula

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

%t Mathematica

P[0, x_] := 729 + 1296*x + 576*x^2;
P[n_, x_] := P[n, x] = Expand[(27 + 24*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] = 729 + 1296*x + 576*x^2; for(n = 1, N, my(i = n + 1); v[i] = (27 + 24*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_A013611 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.