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	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

All 15 rows

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.