A133080 → basis A013609 → basis A013611

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(9 + 48\,x + 64\,x^{2}\right)\,P_{n-2}(x)\)

\(P_{0}(x) = 1\)

\(P_{1}(x) = 4 + 8\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	4	8							12
2	9	48	64						121
3	36	264	640	512					1452
4	81	864	3456	6144	4096				14641
5	324	4104	20736	52224	65536	32768			175692
6	729	11664	77760	276480	552960	589824	262144		1771561
7	2916	52488	404352	1728000	4423680	6782976	5767168	2097152	21258732

All 15 rows

Technical details

Node id: n-3d3c2b58883ecef7 · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-3e9f76b046734869

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
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:     1;
  1:     4,      8;
  2:     9,     48,      64;
  3:    36,    264,     640,      512;
  4:    81,    864,    3456,     6144,      4096;
  5:   324,   4104,   20736,    52224,     65536,     32768;
  6:   729,  11664,   77760,   276480,    552960,    589824,    262144;
  7:  2916,  52488,  404352,  1728000,   4423680,   6782976,   5767168,   2097152;
  8:  6561, 139968, 1306368,  6967296,  23224320,  49545216,  66060288,  50331648,  16777216;
  9: 26244, 612360, 6345216, 38320128, 148635648, 383975424, 660602880, 729808896, 469762048, 134217728;

%F Formula

The row polynomials satisfy P(n,x) = (9+48*x+64*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 8*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 4 + 8*x;
P[n_, x_] := P[n, x] = Expand[(9 + 48*x + 64*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] = 1; v[2] = 4 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = (9 + 48*x + 64*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 basis_map_A013609 of Interpolation operator: Triangle with an even number of zeros in each line followed by 1 or 2 ones.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Interpolation operator: Triangle with an even number of zeros in each line followed by 1 or 2 ones.?
    
    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

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.