A038305 → basis A013610 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(22 + 27\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 484 + 1188\,x + 729\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	484	1188	729								2401
1	10648	39204	48114	19683							117649
2	234256	1149984	2117016	1732104	531441						5764801
3	5153632	31624560	77623920	95265720	58458510	14348907					282475249
4	113379904	834888384	2561589360	4191691680	3858261660	1894055724	387420489				13841287201
5	2494357888	21428801856	78896952288	161380129680	198057431880	145842290748	59662755306	10460353203			678223072849
6	54875873536	538781303808	2314310600448	5680580564736	8714527002720	8556081057216	5250322466928	1841022163728	282429536481		33232930569601
7	1207269217792	13334837269248	65461928412672	187459158636288	345095269307712	423526012332192	346521282817248	182261194209072	55921048223238	7625597484987	1628413597910449

All 15 rows

Technical details

Node id: n-2f306d780dbae501 · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-b7d0fa5aa93aa995

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:             484,             1188,               729;
  1:           10648,            39204,             48114,              19683;
  2:          234256,          1149984,           2117016,            1732104,             531441;
  3:         5153632,         31624560,          77623920,           95265720,           58458510,           14348907;
  4:       113379904,        834888384,        2561589360,         4191691680,         3858261660,         1894055724,          387420489;
  5:      2494357888,      21428801856,       78896952288,       161380129680,       198057431880,       145842290748,        59662755306,        10460353203;
  6:     54875873536,     538781303808,     2314310600448,      5680580564736,      8714527002720,      8556081057216,      5250322466928,      1841022163728,       282429536481;
  7:   1207269217792,   13334837269248,    65461928412672,    187459158636288,    345095269307712,    423526012332192,    346521282817248,    182261194209072,     55921048223238,      7625597484987;
  8:  26559922791424,  325962688803840,  1800203031348480,   5891573557140480,  12653493207949440,  18635144542616448,  19058670554948640,  13365820908665280,   6151315304556180,   1677631446697140,   205891132094649;
  9: 584318301411328, 7888297069052928, 48405459287370240, 178220100103499520, 437449336617680640, 751617496552196736, 922439654859514176, 808632164974249440, 496206101234198520, 202993405050353940, 49825653966905058, 5559060566555523;

%F Formula

The row polynomials satisfy P(n,x) = (22+27*x)*P(n-1,x) for n >= 1, with P(0,x) = 484 + 1188*x + 729*x^2.

%t Mathematica

P[0, x_] := 484 + 1188*x + 729*x^2;
P[n_, x_] := P[n, x] = Expand[(22 + 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] = 484 + 1188*x + 729*x^2; for(n = 1, N, my(i = n + 1); v[i] = (22 + 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)*10^(i-j)*3^j.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle whose (i,j)-th entry is binomial(i,j)*10^(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

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.