A133080 → basis A013609 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(9 + 36\,x + 36\,x^{2}\right)\,P_{n-2}(x)\)

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

\(P_{1}(x) = 4 + 6\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	4	6														10
2	9	36	36													81
3	36	198	360	216												810
4	81	648	1944	2592	1296											6561
5	324	3078	11664	22032	20736	7776										65610
6	729	8748	43740	116640	174960	139968	46656									531441
7	2916	39366	227448	729000	1399680	1609632	1026432	279936								5314410
8	6561	104976	734832	2939328	7348320	11757312	11757312	6718464	1679616							43046721
9	26244	459270	3569184	16166304	47029248	91119168	117573120	97417728	47029248	10077696						430467210
10	59049	1180980	10628820	56687040	198404640	476171136	793618560	906992640	680244480	302330880	60466176					3486784401
11	236196	5078214	49601160	290521080	1133740800	3095112384	6031501056	8389681920	8162933760	5290790400	2055849984	362797056				34867844010
12	531441	12754584	140300424	935336160	4209012720	13468840704	31427294976	53875362816	67344203520	59861514240	35916908544	13060694016	2176782336			282429536481
13	2125764	54206982	637729200	4583147184	22448067840	79129439136	206522224128	404065221120	592628990976	643511278080	502836719616	267744227328	87071293440	13060694016		2824295364810
14	4782969	133923132	1741000716	13928005728	76604031504	306416126016	919248378048	2101139149824	3676993512192	4902658016256	4902658016256	3565569466368	1782784733184	548549148672	78364164096	22876792454961

Technical details

Node id: n-7f72626fdabfb38d · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-0fc72d33f55fbdc4

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,      6;
  2:     9,     36,      36;
  3:    36,    198,     360,      216;
  4:    81,    648,    1944,     2592,     1296;
  5:   324,   3078,   11664,    22032,    20736,     7776;
  6:   729,   8748,   43740,   116640,   174960,   139968,     46656;
  7:  2916,  39366,  227448,   729000,  1399680,  1609632,   1026432,   279936;
  8:  6561, 104976,  734832,  2939328,  7348320, 11757312,  11757312,  6718464,  1679616;
  9: 26244, 459270, 3569184, 16166304, 47029248, 91119168, 117573120, 97417728, 47029248, 10077696;

%F Formula

The row polynomials satisfy P(n,x) = (9+36*x+36*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 6*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 4 + 6*x;
P[n_, x_] := P[n, x] = Expand[(9 + 36*x + 36*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 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = (9 + 36*x + 36*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

Related ?

Same fingerprint ?

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.