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 ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
040010807292209
18000324004374019683103823
2160000864000174960015746405314414879681
332000002160000058320000787320005314410014348907229345007
464000000518400000174960000031492800003188646000172186884038742048910779215329
5128000000012096000000489888000001102248000001488034800001205308188005423886846010460353203506623120463
62560000000027648000000013063680000003527193600000595213920000064283103360004339109476800167365651248028242953648123811286661761
75120000000006220800000000335923200000001058158080000002142770112000002892739651200002603465686080001506290861232005083731656658076255974849871119130473102767

All 15 rows

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.