A038321 → basis A013610 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(39 + 42\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1521 + 3276\,x + 1764\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	1521	3276	1764								6561
1	59319	191646	206388	74088							531441
2	2313441	9965592	16098264	11557728	3111696						43046721
3	90224199	485822610	1046387160	1126878480	606780720	130691232					3486784401
4	3518743761	22736498148	61213648860	87896521440	70993344240	30581748288	5489031744				282429536481
5	137231006679	1034510665734	3342265227756	5998937588280	6460394325840	4174408641312	1498505666112	230539333248			22876792454961
6	5352009260481	46109618244144	173797791843312	374333705508672	503910757415520	434138498696448	233766883913472	71928271973376	9682651996416		1853020188851841
7	208728361158759	2023059500461818	8714717848143216	21898521772257312	35374535170569504	38095653260613312	27350725417876224	12623411731327488	3398610850742016	406671383849472	150094635296999121

All 15 rows

Technical details

Node id: n-23763ec4ded1b991 · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-b21306f08ea8390f

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
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:               1521,                3276,                 1764;
  1:              59319,              191646,               206388,                74088;
  2:            2313441,             9965592,             16098264,             11557728,               3111696;
  3:           90224199,           485822610,           1046387160,           1126878480,             606780720,             130691232;
  4:         3518743761,         22736498148,          61213648860,          87896521440,           70993344240,           30581748288,            5489031744;
  5:       137231006679,       1034510665734,        3342265227756,        5998937588280,         6460394325840,         4174408641312,         1498505666112,          230539333248;
  6:      5352009260481,      46109618244144,      173797791843312,      374333705508672,       503910757415520,       434138498696448,       233766883913472,        71928271973376,        9682651996416;
  7:    208728361158759,    2023059500461818,     8714717848143216,    21898521772257312,     35374535170569504,     38095653260613312,     27350725417876224,     12623411731327488,     3398610850742016,      406671383849472;
  8:   8140406085191601,   87665911686678780,   424842495096981780,  1220060498740050240,   2299344786087017760,   2971460954327838336,   2666695728242931840,   1641043525072573440,   662729115894693120,   158601839701294080,   17080198121677824;
  9: 317475837322472439, 3760867611358519662, 20250825599622798180, 65425744244935194120, 140916987604475802720, 212459458234440441024, 228802493483243551872, 176001918064033501440, 94770263572941116160, 34020094615927580160, 7327404994199786496, 717368321110468608;

%F Formula

The row polynomials satisfy P(n,x) = (39+42*x)*P(n-1,x) for n >= 1, with P(0,x) = 1521 + 3276*x + 1764*x^2.

%t Mathematica

P[0, x_] := 1521 + 3276*x + 1764*x^2;
P[n_, x_] := P[n, x] = Expand[(39 + 42*x)*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] = 1521 + 3276*x + 1764*x^2; for(n = 1, N, my(i = n + 1); v[i] = (39 + 42*x)*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 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_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.?
    
    Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    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 ?

A038327 → x ↦ x+1 → x ↦ x/(x+1)

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
[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.