A038331 → basis A013609 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(27 + 20\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 729 + 1080\,x + 400\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	729	1080	400								2209
1	19683	43740	32400	8000							103823
2	531441	1574640	1749600	864000	160000						4879681
3	14348907	53144100	78732000	58320000	21600000	3200000					229345007
4	387420489	1721868840	3188646000	3149280000	1749600000	518400000	64000000				10779215329
5	10460353203	54238868460	120530818800	148803480000	110224800000	48988800000	12096000000	1280000000			506623120463
6	282429536481	1673656512480	4339109476800	6428310336000	5952139200000	3527193600000	1306368000000	276480000000	25600000000		23811286661761
7	7625597484987	50837316566580	150629086123200	260346568608000	289273965120000	214277011200000	105815808000000	33592320000000	6220800000000	512000000000	1119130473102767

All 15 rows

Technical details

Node id: n-4202192cb3f33bc8 · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-94e7d97c3019c560

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,              1080,                400;
  1:            19683,             43740,              32400,               8000;
  2:           531441,           1574640,            1749600,             864000,             160000;
  3:         14348907,          53144100,           78732000,           58320000,           21600000,            3200000;
  4:        387420489,        1721868840,         3188646000,         3149280000,         1749600000,          518400000,           64000000;
  5:      10460353203,       54238868460,       120530818800,       148803480000,       110224800000,        48988800000,        12096000000,         1280000000;
  6:     282429536481,     1673656512480,      4339109476800,      6428310336000,      5952139200000,      3527193600000,      1306368000000,       276480000000,       25600000000;
  7:    7625597484987,    50837316566580,    150629086123200,    260346568608000,    289273965120000,    214277011200000,    105815808000000,     33592320000000,     6220800000000,      512000000000;
  8:  205891132094649,  1525119496997400,   5083731656658000,  10041939074880000,  13017328430400000,  11570958604800000,   7142567040000000,   3023308800000000,   839808000000000,   138240000000000,   10240000000000;
  9: 5559060566555523, 45296049060822780, 167763144669714000, 372806988154920000, 552306649118400000, 572762450937600000, 424268482176000000, 224480678400000000, 83140992000000000, 20528640000000000, 3041280000000000, 204800000000000;

%F Formula

The row polynomials satisfy P(n,x) = (27+20*x)*P(n-1,x) for n >= 1, with P(0,x) = 729 + 1080*x + 400*x^2.

%t Mathematica

P[0, x_] := 729 + 1080*x + 400*x^2;
P[n_, x_] := P[n, x] = Expand[(27 + 20*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 + 1080*x + 400*x^2; for(n = 1, N, my(i = n + 1); v[i] = (27 + 20*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.