Real-rooted atlas

A038301 → reverse → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(20 + 27\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 400 + 1080\,x + 729\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
040010807292209
18000324004374019683103823
2160000864000174960015746405314414879681
332000002160000058320000787320005314410014348907229345007
464000000518400000174960000031492800003188646000172186884038742048910779215329
5128000000012096000000489888000001102248000001488034800001205308188005423886846010460353203506623120463
62560000000027648000000013063680000003527193600000595213920000064283103360004339109476800167365651248028242953648123811286661761
75120000000006220800000000335923200000001058158080000002142770112000002892739651200002603465686080001506290861232005083731656658076255974849871119130473102767
810240000000000138240000000000839808000000000302330880000000071425670400000001157095860480000013017328430400000100419390748800005083731656658000152511949699740020589113209464952599132235830049
9204800000000000304128000000000020528640000000000831409920000000002244806784000000004242684821760000005727624509376000005523066491184000003728069881549200001677631446697140004529604906082278055590605665555232472159215084012303
1040960000000000006635520000000000049268736000000000022170931200000000006734420352000000000145463479603200000002291049803750400000026510719157683200000223684192892952000001342105157357712000054355258872987336001334174535973325520150094635296999121116191483108948578241
118192000000000000014376960000000000001164533760000000000057644421120000000000194549921280000000000472756308710400000000850961355678720000000114879783016627200000011631578030433504000008723683522825128000004710789102325569120001734426896765323176003902460517721977146040525551530189762675460999706120583177327
12163840000000000000030965760000000000000271724544000000000000146731253760000000000054473977958400000000001470797404876800000000029783647448755200000000459519132066508800000005428069747535635200000048852627727820716800000329755237162789838400001618798436980968297600054634447248107680044001134715442845313354760109418989131512359209256666986187667409334369
13327680000000000000006635520000000000000006270566400000000000000366828134400000000000001485653944320000000000004412392214630400000000009927882482918400000000001723196745249408000000000232631560608670080000000024426313863910358400000001978531422976739030400000121409882773572622320000054634447248107680044000017020731642679700321400032825696739453707762700295431270655083369864312063348350820368238715343
1465536000000000000000014155776000000000000000143327232000000000000000902961561600000000000000396174385152000000000000012836050078924800000000000317692239453388800000000006126921760886784000000000093052624243468032000000000111663149092161638400000000105521675892092748288000000777023249750864782848000004370755779848614403520000018155447085525013676160000525211147831259324203200094538006609626678356576079766443076872509863361566977372488557307219621121
Technical details
Node id
n-c380fc10621a239e · revision 0 · JSON
Definition source
derived:basis_map_A013610
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:             400,             1080,               729;
  1:            8000,            32400,             43740,             19683;
  2:          160000,           864000,           1749600,           1574640,             531441;
  3:         3200000,         21600000,          58320000,          78732000,           53144100,           14348907;
  4:        64000000,        518400000,        1749600000,        3149280000,         3188646000,         1721868840,          387420489;
  5:      1280000000,      12096000000,       48988800000,      110224800000,       148803480000,       120530818800,        54238868460,        10460353203;
  6:     25600000000,     276480000000,     1306368000000,     3527193600000,      5952139200000,      6428310336000,      4339109476800,      1673656512480,       282429536481;
  7:    512000000000,    6220800000000,    33592320000000,   105815808000000,    214277011200000,    289273965120000,    260346568608000,    150629086123200,     50837316566580,      7625597484987;
  8:  10240000000000,  138240000000000,   839808000000000,  3023308800000000,   7142567040000000,  11570958604800000,  13017328430400000,  10041939074880000,   5083731656658000,   1525119496997400,   205891132094649;
  9: 204800000000000, 3041280000000000, 20528640000000000, 83140992000000000, 224480678400000000, 424268482176000000, 572762450937600000, 552306649118400000, 372806988154920000, 167763144669714000, 45296049060822780, 5559060566555523;
%F Formula
The row polynomials satisfy P(n,x) = (20+27*x)*P(n-1,x) for n >= 1, with P(0,x) = 400 + 1080*x + 729*x^2.
%t Mathematica
P[0, x_] := 400 + 1080*x + 729*x^2;
P[n_, x_] := P[n, x] = Expand[(20 + 27*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] = 400 + 1080*x + 729*x^2; for(n = 1, N, my(i = n + 1); v[i] = (20 + 27*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

Proof chains ? JSON

Same fingerprint ?

References

  1. [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
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.