A103633 → basis A013609 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(12 + 28\,x + 16\,x^{2}\right)\,P_{n-2}(x)\)

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

\(P_{1}(x) = 3 + 4\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	3	4														7
2	12	28	16													56
3	36	132	160	64												392
4	144	672	1168	896	256											3136
5	432	2592	6192	7360	4352	1024										21952
6	1728	12096	35136	54208	46848	21504	4096									175616
7	5184	43200	153792	303168	357376	251904	98304	16384								1229312
8	20736	193536	787968	1827840	2642176	2437120	1400832	458752	65536							9834496
9	62208	663552	3138048	8635392	15237888	17880064	13950976	6979584	2031616	262144						68841472
10	248832	2903040	15206400	47093760	95493120	132471808	127324160	83722240	36044800	9175040	1048576					550731776
11	746496	9704448	57231360	202106880	474854400	779387904	911859712	760463360	443023360	171704320	39845888	4194304				3855122432
12	2985984	41803776	267743232	1037352960	2707845120	5016969216	6764990464	6689292288	4813946880	2458910720	846200832	176160768	16777216			30840979456
13	8957952	137355264	970444800	4183031808	12272947200	25882288128	40362848256	47127838720	41199009792	26632519680	12374245376	3913285632	754974720	67108864		215886856192
14	35831808	585252864	4431200256	20613906432	65823916032	152620941312	264980545536	349962747904	353307394048	271326117888	156027060224	65150124032	18673041408	3288334336	268435456	1727094849536

Technical details

Node id: n-20d13d43b8d2496f · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-409ecd303d9e1d72

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:     3,      4;
  2:    12,     28,      16;
  3:    36,    132,     160,      64;
  4:   144,    672,    1168,     896,      256;
  5:   432,   2592,    6192,    7360,     4352,     1024;
  6:  1728,  12096,   35136,   54208,    46848,    21504,     4096;
  7:  5184,  43200,  153792,  303168,   357376,   251904,    98304,   16384;
  8: 20736, 193536,  787968, 1827840,  2642176,  2437120,  1400832,  458752,   65536;
  9: 62208, 663552, 3138048, 8635392, 15237888, 17880064, 13950976, 6979584, 2031616, 262144;

%F Formula

The row polynomials satisfy P(n,x) = (12+28*x+16*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 4*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 4*x;
P[n_, x_] := P[n, x] = Expand[(12 + 28*x + 16*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] = 3 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (12 + 28*x + 16*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 Triangle read by rows: triangle of repeated stepped binomial coefficients.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows: triangle of repeated stepped binomial coefficients.?
    
    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

Related ?

Same fingerprint ?

A064861 → basis A013609

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.