A272867 → basis A013611 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(16 + 24\,x + 6\,x^{2}\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	16	24	6													46
2	256	768	768	288	36											2116
3	4096	18432	32256	27648	12096	2592	216									97336
4	65536	393216	983040	1327104	1050624	497664	138240	20736	1296							4477456
5	1048576	7864320	25559040	47185920	54558720	41140224	20459520	6635520	1347840	155520	7776					205962976
6	16777216	150994944	603979776	1415577600	2158755840	2250768384	1642070016	844038144	303575040	74649600	11943936	1119744	46656			9474296896
7	268435456	2818572288	13388218368	38050725888	72137834496	96315899904	93244096512	66418900992	34966536192	13544423424	3804143616	752467968	99283968	7838208	279936	435817657216

All 15 rows

Technical details

Node id: n-524d4090108719a6 · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-cb1c90404b4cec63

Observed on the stored rows ?

Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:           1;
  1:          16,           24,             6;
  2:         256,          768,           768,            288,             36;
  3:        4096,        18432,         32256,          27648,          12096,            2592,             216;
  4:       65536,       393216,        983040,        1327104,        1050624,          497664,          138240,           20736,            1296;
  5:     1048576,      7864320,      25559040,       47185920,       54558720,        41140224,        20459520,         6635520,         1347840,          155520,           7776;
  6:    16777216,    150994944,     603979776,     1415577600,     2158755840,      2250768384,      1642070016,       844038144,       303575040,        74649600,       11943936,        1119744,         46656;
  7:   268435456,   2818572288,   13388218368,    38050725888,    72137834496,     96315899904,     93244096512,     66418900992,     34966536192,     13544423424,     3804143616,      752467968,      99283968,       7838208,       279936;
  8:  4294967296,  51539607552,  283467841536,   947040288768,  2147752083456,   3500666781696,   4236314148864,   3878456131584,   2712982781952,   1454421049344,   595731677184,   184605474816,   42472636416,    7023034368,    788299776,    53747712,    1679616;
  9: 68719476736, 927712935936, 5798205849600, 22265110462464, 58793807314944, 113238960242688, 164683541643264, 184730838368256, 161908556562432, 111653060345856, 60715708710912, 25977774145536, 8684483641344, 2239344672768, 436001439744, 61917364224, 6046617600, 362797056, 10077696;

%F Formula

The row polynomials satisfy P(n,x) = (16+24*x+6*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(16 + 24*x + 6*x^2)*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (16 + 24*x + 6*x^2)*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 Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from basis_map_A013611 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    ↑ Iterated real-rootedness preservers (above)
    
    Parent-row provers of polytool polytool ? step › direct check
    
    ↑ Parent-row provers of polytool (above)

Related ?

Same fingerprint ?

A272867 → basis A013611

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.