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	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	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
8	205891132094649	1525119496997400	5083731656658000	10041939074880000	13017328430400000	11570958604800000	7142567040000000	3023308800000000	839808000000000	138240000000000	10240000000000							52599132235830049
9	5559060566555523	45296049060822780	167763144669714000	372806988154920000	552306649118400000	572762450937600000	424268482176000000	224480678400000000	83140992000000000	20528640000000000	3041280000000000	204800000000000						2472159215084012303
10	150094635296999121	1334174535973325520	5435525887298733600	13421051573577120000	22368419289295200000	26510719157683200000	22910498037504000000	14546347960320000000	6734420352000000000	2217093120000000000	492687360000000000	66355200000000000	4096000000000000					116191483108948578241
11	4052555153018976267	39024605177219771460	173442689676532317600	471078910232556912000	872368352282512800000	1163157803043350400000	1148797830166272000000	850961355678720000000	472756308710400000000	194549921280000000000	57644421120000000000	11645337600000000000	1437696000000000000	81920000000000000				5460999706120583177327
12	109418989131512359209	1134715442845313354760	5463444724810768004400	16187984369809682976000	32975523716278983840000	48852627727820716800000	54280697475356352000000	45951913206650880000000	29783647448755200000000	14707974048768000000000	5447397795840000000000	1467312537600000000000	271724544000000000000	30965760000000000000	1638400000000000000			256666986187667409334369
13	2954312706550833698643	32825696739453707762700	170207316426797003214000	546344472481076800440000	1214098827735726223200000	1978531422976739030400000	2442631386391035840000000	2326315606086700800000000	1723196745249408000000000	992788248291840000000000	441239221463040000000000	148565394432000000000000	36682813440000000000000	6270566400000000000000	663552000000000000000	32768000000000000000		12063348350820368238715343
14	79766443076872509863361	945380066096266783565760	5252111478312593242032000	18155447085525013676160000	43707557798486144035200000	77702324975086478284800000	105521675892092748288000000	111663149092161638400000000	93052624243468032000000000	61269217608867840000000000	31769223945338880000000000	12836050078924800000000000	3961743851520000000000000	902961561600000000000000	143327232000000000000000	14155776000000000000000	655360000000000000000	566977372488557307219621121

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.