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	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	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
8	205891132094649	2287679245496100	11438396227480500	33891544377720000	65900225178900000	87866966905200000	81358302690000000	51656065200000000	21523360500000000	5314410000000000	590490000000000							362033331456891249
9	5559060566555523	67944073591234170	377467075506856500	1258223585022855000	2796052411161900000	4349414861807400000	4832683179786000000	3835462841100000000	2130812689500000000	789189885000000000	175375530000000000	17714700000000000						20635899893042801193
10	150094635296999121	2001261803959988280	12229933246422150600	45296049060822780000	113240122652056950000	201315773603656800000	260964891708444000000	248537992103280000000	172595827849500000000	85232507580000000000	28410835860000000000	5739562800000000000	531441000000000000					1176246293903439668001
11	4052555153018976267	58536907765829657190	390246051772197714600	1589891322034879578000	4416364783430221050000	8832729566860442100000	13085525284237692000000	14539472538041880000000	12116227115034900000000	7479152540145000000000	3324067795620000000000	1007293271400000000000	186535791000000000000	15943230000000000000				67046038752496061076057
12	109418989131512359209	1702073164267970032140	12292750630824228009900	54634447248107680044000	166938588813662355690000	370974641808138568200000	618291069680230947000000	785131517054261520000000	763322308247198700000000	565423932034962000000000	314124406686090000000000	126918952196400000000000	35255264499000000000000	6026540940000000000000	478296900000000000000			3821624208892275481335249
13	2954312706550833698643	49238545109180561644050	382966461960293257231500	1843912594623634201485000	6146375315412114004950000	15024472993229612012100000	27823098135610392615000000	39747283050871989450000000	44163647834302210500000000	38166115412359935000000000	25444076941573290000000000	12850543909885500000000000	4759460707365000000000000	1220374540350000000000000	193710244500000000000000	14348907000000000000000		217832579906859702436109193
14	79766443076872509863361	1418070099144400175348640	11817250826203334794572000	61274633913646921157040000	221269511354836104178200000	590052030279562944475200000	1201957839458368960968000000	1907869586441855493600000000	2384836983052319367000000000	2355394551162784560000000000	1831973539793276880000000000	1110286993814107200000000000	514021756395420000000000000	175733933810400000000000000	41841412812000000000000000	6198727824000000000000000	430467210000000000000000	12416457054691003038858224001

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.