A272866 → basis A013610 → basis A007318

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one sign Hadamard kernel

Recurrence

\(P_{n}(x) = \left(29 + 33\,x + 9\,x^{2}\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	29	33	9													71
2	841	1914	1611	594	81											5041
3	24389	83259	117450	87615	36450	8019	729									357911
4	707281	3219348	6373098	7166016	5005395	2223936	613818	96228	6561							25411681
5	20511149	116701365	297423855	447100830	438992865	294166323	136239165	43062030	8890155	1082565	59049					1804229351
6	594823321	4061207502	12661037181	23831223570	30161935170	27041495382	17609360229	8392188222	2905013970	712327770	117448461	11691702	531441			128100283921
7	17249876309	137404187151	506543335704	1145470578021	1775075832369	1994028238818	1674498210777	1067855804433	519671858517	192052660338	53057947509	10625808501	1458274104	122762871	4782969	9095120158391

All 15 rows

Technical details

Node id: n-6a3923af89150a19 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-40d39f31cccb6263

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:             29,              33,               9;
  2:            841,            1914,            1611,              594,               81;
  3:          24389,           83259,          117450,            87615,            36450,             8019,              729;
  4:         707281,         3219348,         6373098,          7166016,          5005395,          2223936,           613818,            96228,             6561;
  5:       20511149,       116701365,       297423855,        447100830,        438992865,        294166323,        136239165,         43062030,          8890155,          1082565,            59049;
  6:      594823321,      4061207502,     12661037181,      23831223570,      30161935170,      27041495382,      17609360229,       8392188222,       2905013970,        712327770,        117448461,        11691702,          531441;
  7:    17249876309,    137404187151,    506543335704,    1145470578021,    1775075832369,    1994028238818,    1674498210777,    1067855804433,     519671858517,     192052660338,      53057947509,     10625808501,      1458274104,      122762871,        4782969;
  8:   500246412961,   4553967345576,  19379343798180,   51171214525200,   93836618234730,  126713556596088,  130339062484848,  104172513433560,   65380209340275,   32329400720760,   12553464995568,   3787534657368,    870463157130,   147315445200,    17314347780,    1262703816,     43046721;
  9: 14507145975869, 148573184649417, 716784110267877, 2164469272680924, 4584326102322390, 7231842473759442, 8805909743844066, 8462613960938016, 6506770576539087, 4032652150033155, 2019342592719027, 815067456404256, 263213260210026, 67085244012402, 13197694191390, 1933830894204, 198746710857, 12784876137, 387420489;

%F Formula

The row polynomials satisfy P(n,x) = (29+33*x+9*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[(29 + 33*x + 9*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] = (29 + 33*x + 9*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 (−∞, -1]
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 Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from basis_map_A013610 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) 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
    
    Parent-row provers of polytool polytool ? step › direct check
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Parent-row provers of polytool (above)
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from basis_map_A013610 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and…?
  - ↑ Geometric basis map (elementary) (above)

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.