Real-rooted atlas

A038323 → basis A013610 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(27 + 20\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1272047
272910804002209
31968343740324008000103823
4531441157464017496008640001600004879681
514348907531441007873200058320000216000003200000229345007
638742048917218688403188646000314928000017496000005184000006400000010779215329
7104603532035423886846012053081880014880348000011022480000048988800000120960000001280000000506623120463
82824295364811673656512480433910947680064283103360005952139200000352719360000013063680000002764800000002560000000023811286661761
97625597484987508373165665801506290861232002603465686080002892739651200002142770112000001058158080000003359232000000062208000000005120000000001119130473102767
1020589113209464915251194969974005083731656658000100419390748800001301732843040000011570958604800000714256704000000030233088000000008398080000000001382400000000001024000000000052599132235830049
11555906056655552345296049060822780167763144669714000372806988154920000552306649118400000572762450937600000424268482176000000224480678400000000831409920000000002052864000000000030412800000000002048000000000002472159215084012303
1215009463529699912113341745359733255205435525887298733600134210515735771200002236841928929520000026510719157683200000229104980375040000001454634796032000000067344203520000000002217093120000000000492687360000000000663552000000000004096000000000000116191483108948578241
134052555153018976267390246051772197714601734426896765323176004710789102325569120008723683522825128000001163157803043350400000114879783016627200000085096135567872000000047275630871040000000019454992128000000000057644421120000000000116453376000000000001437696000000000000819200000000000005460999706120583177327
14109418989131512359209113471544284531335476054634447248107680044001618798436980968297600032975523716278983840000488526277278207168000005428069747535635200000045951913206650880000000297836474487552000000001470797404876800000000054473977958400000000001467312537600000000000271724544000000000000309657600000000000001638400000000000000256666986187667409334369
Technical details
Node id
n-27bd83a6a2318024 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-94e7d97c3019c560

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • 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:             1;
  1:            27,             20;
  2:           729,           1080,             400;
  3:         19683,          43740,           32400,            8000;
  4:        531441,        1574640,         1749600,          864000,          160000;
  5:      14348907,       53144100,        78732000,        58320000,        21600000,         3200000;
  6:     387420489,     1721868840,      3188646000,      3149280000,      1749600000,       518400000,        64000000;
  7:   10460353203,    54238868460,    120530818800,    148803480000,    110224800000,     48988800000,     12096000000,     1280000000;
  8:  282429536481,  1673656512480,   4339109476800,   6428310336000,   5952139200000,   3527193600000,   1306368000000,   276480000000,   25600000000;
  9: 7625597484987, 50837316566580, 150629086123200, 260346568608000, 289273965120000, 214277011200000, 105815808000000, 33592320000000, 6220800000000, 512000000000;
%F Formula
The row polynomials satisfy P(n,x) = (27+20*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
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] = 1; 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

Proof chains ? JSON

Same fingerprint ?

References

  1. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [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
  4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  5. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  6. [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
  7. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.