A391533 → Pₙ + Pₙ₊₁ → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(2 + 3\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 27 + 81\,x + 81\,x^{2} + 27\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	x¹⁷	P_n(1)
0	27	81	81	27															216
1	54	243	405	297	81														1080
2	108	648	1539	1809	1053	243													5400
3	216	1620	5022	8235	7533	3645	729												27000
4	432	3888	14904	31536	39771	29889	12393	2187											135000
5	864	9072	41472	107784	174150	179091	114453	41553	6561										675000
6	1728	20736	110160	339984	671652	880632	766179	426465	137781	19683									3375000
7	3456	46656	282528	1010448	2363256	3776220	4174254	3151467	1554957	452709	59049								16875000
8	6912	103680	705024	2868480	7757856	14642208	19677168	18825696	12564315	5570289	1476225	177147							84375000
9	13824	228096	1721088	7852032	24121152	52557984	83280960	96682896	81605718	48833523	19663317	4782969	531441						421875000
10	27648	497664	4126464	20867328	71798400	177479424	324235872	443208672	453260124	342484200	185827203	68555889	15411789	1594323					2109375000
11	55296	1078272	9745920	54114048	206198784	570354048	1180910016	1859124960	2236146264	2044748772	1399107006	694593387	236491245	49424013	4782969				10546875000
12	110592	2322432	22726656	137465856	574739712	1759304448	4072882176	7260979968	10049667408	10797936336	8932460328	5586507792	2556762651	808321761	157837977	14348907			52734375000
13	221184	4976640	52420608	343111680	1561876992	5242828032	13423677696	26740606464	41882274720	51744874896	50258729664	37970396568	21873048678	9286931475	2740641237	502211745	43046721		263671875000
14	442368	10616832	119771136	843485184	4153089024	15171287040	42575839488	93752246016	163986368832	229136573952	255752084016	226716982128	157657287060	84193008984	33342076899	9226347201	1592728677	129140163	1318359375000

Technical details

Node id: n-2342641336502fe1 · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-6d253706f4f62393

Observed on the stored rows ?

Degrees: 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 (affine: slope 1, intercept 3)
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:    27,     81,      81,      27;
  1:    54,    243,     405,     297,       81;
  2:   108,    648,    1539,    1809,     1053,      243;
  3:   216,   1620,    5022,    8235,     7533,     3645,      729;
  4:   432,   3888,   14904,   31536,    39771,    29889,    12393,     2187;
  5:   864,   9072,   41472,  107784,   174150,   179091,   114453,    41553,     6561;
  6:  1728,  20736,  110160,  339984,   671652,   880632,   766179,   426465,   137781,    19683;
  7:  3456,  46656,  282528, 1010448,  2363256,  3776220,  4174254,  3151467,  1554957,   452709,    59049;
  8:  6912, 103680,  705024, 2868480,  7757856, 14642208, 19677168, 18825696, 12564315,  5570289,  1476225,  177147;
  9: 13824, 228096, 1721088, 7852032, 24121152, 52557984, 83280960, 96682896, 81605718, 48833523, 19663317, 4782969, 531441;

%F Formula

The row polynomials satisfy P(n,x) = (2+3*x)*P(n-1,x) for n >= 1, with P(0,x) = 27 + 81*x + 81*x^2 + 27*x^3.

%t Mathematica

P[0, x_] := 27 + 81*x + 81*x^2 + 27*x^3;
P[n_, x_] := P[n, x] = Expand[(2 + 3*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] = 27 + 81*x + 81*x^2 + 27*x^3; for(n = 1, N, my(i = n + 1); v[i] = (2 + 3*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 shift_x_plus_one of Riordan array ((1+x)^3,x).?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Riordan array ((1+x)^3,x).?
    
    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
  - Geometric basis map (elementary) polytool ? step ›
    
    from Riordan array ((1+x)^3,x).?
    
    ↑ Row-hypergeometric rows (multiplier sequences) (above)
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (above)

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