Veronese parent of A109447 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(4 + 12\,x + 9\,x^{2}\right)\,P_{n-2}(x)\)

\(P_{0}(x) = 2 + 9\,x + 9\,x^{2}\)

\(P_{1}(x) = 4 + 12\,x + 9\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	P_n(1)
0	2	9	9							20
1	4	12	9							25
2	8	60	162	189	81					500
3	16	96	216	216	81					625
4	32	336	1440	3240	4050	2673	729			12500
5	64	576	2160	4320	4860	2916	729			15625
6	128	1728	10080	33264	68040	88452	71442	32805	6561	312500
7	256	3072	16128	48384	90720	108864	81648	34992	6561	390625

All 15 rows

Technical details

Node id: n-2c17d6db73d0be9b · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-6cf66d9918c78db1

Observed on the stored rows ?

Degrees: 2, 2, 4, 4, 6, 6, 8, 8, 10, 10, 12, 12, 14, 14, 16 (irregular)
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:    2,     9,      9;
  1:    4,    12,      9;
  2:    8,    60,    162,    189,      81;
  3:   16,    96,    216,    216,      81;
  4:   32,   336,   1440,   3240,    4050,    2673,     729;
  5:   64,   576,   2160,   4320,    4860,    2916,     729;
  6:  128,  1728,  10080,  33264,   68040,   88452,   71442,   32805,    6561;
  7:  256,  3072,  16128,  48384,   90720,  108864,   81648,   34992,    6561;
  8:  512,  8448,  62208, 269568,  762048, 1469664, 1959552, 1784592, 1062882, 373977, 59049;
  9: 1024, 15360, 103680, 414720, 1088640, 1959552, 2449440, 2099520, 1180980, 393660, 59049;

%F Formula

The row polynomials satisfy P(n,x) = (4+12*x+9*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 2 + 9*x + 9*x^2, P(1,x) = 4 + 12*x + 9*x^2.

%t Mathematica

P[0, x_] := 2 + 9*x + 9*x^2;
P[1, x_] := 4 + 12*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[(4 + 12*x + 9*x^2)*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 2 + 9*x + 9*x^2; v[2] = 4 + 12*x + 9*x^2; for(n = 2, N, my(i = n + 1); v[i] = (4 + 12*x + 9*x^2)*v[i - 2]); 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 veronese parent of Binomial coefficients C(n,k) with n-k odd, read by rows.?
  - 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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.