A038331 → basis A013609 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(27 + 30\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 729 + 1620\,x + 900\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	729	1620	900								3249
1	19683	65610	72900	27000							185193
2	531441	2361960	3936600	2916000	810000						10556001
3	14348907	79716150	177147000	196830000	109350000	24300000					601692057
4	387420489	2582803260	7174453500	10628820000	8857350000	3936600000	729000000				34296447249
5	10460353203	81358302690	271194342300	502211745000	558013050000	372008700000	137781000000	21870000000			1954897493193
6	282429536481	2510484768720	9762996322800	21695547384000	30132704700000	26784626400000	14880348000000	4723920000000	656100000000		111429157112001
7	7625597484987	76255974849870	338915443777200	878669669052000	1464449448420000	1627166053800000	1205308188000000	573956280000000	159432300000000	19683000000000	6351461955384057

All 15 rows

Technical details

Node id: n-fbce10918920af94 · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-249d425e0def4c4f

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:              729,              1620,                900;
  1:            19683,             65610,              72900,               27000;
  2:           531441,           2361960,            3936600,             2916000,              810000;
  3:         14348907,          79716150,          177147000,           196830000,           109350000,            24300000;
  4:        387420489,        2582803260,         7174453500,         10628820000,          8857350000,          3936600000,           729000000;
  5:      10460353203,       81358302690,       271194342300,        502211745000,        558013050000,        372008700000,        137781000000,         21870000000;
  6:     282429536481,     2510484768720,      9762996322800,      21695547384000,      30132704700000,      26784626400000,      14880348000000,       4723920000000,        656100000000;
  7:    7625597484987,    76255974849870,    338915443777200,     878669669052000,    1464449448420000,    1627166053800000,    1205308188000000,     573956280000000,     159432300000000,     19683000000000;
  8:  205891132094649,  2287679245496100,  11438396227480500,   33891544377720000,   65900225178900000,   87866966905200000,   81358302690000000,   51656065200000000,   21523360500000000,   5314410000000000,    590490000000000;
  9: 5559060566555523, 67944073591234170, 377467075506856500, 1258223585022855000, 2796052411161900000, 4349414861807400000, 4832683179786000000, 3835462841100000000, 2130812689500000000, 789189885000000000, 175375530000000000, 17714700000000000;

%F Formula

The row polynomials satisfy P(n,x) = (27+30*x)*P(n-1,x) for n >= 1, with P(0,x) = 729 + 1620*x + 900*x^2.

%t Mathematica

P[0, x_] := 729 + 1620*x + 900*x^2;
P[n_, x_] := P[n, x] = Expand[(27 + 30*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] = 729 + 1620*x + 900*x^2; for(n = 1, N, my(i = n + 1); v[i] = (27 + 30*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_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^j.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^j.?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    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
    
    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
[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.