Real-rooted atlas

A038323 → basis A013610 → basis A007318

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(47 + 27\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1472774
2220925387295476
310382317892910278919683405224
44879681112128849662166370040453144129986576
5229345007658756935756869670434797470124888635143489072219006624
610779215329371538911345335931173540870962180176092975354046391774387420489164206490176
750662312046320372716971813511042712163336163663930519311529630056656314468231274613408811046035320312151280273024
82381128666176110943059402000822002534329554825279507527573618152837852247083425808001816239627320856283933092804328282429536481899194740203776
91119130473102767578614265880792313295817173430972178220528069393881535730082300096288222791961920423378745224073548831849128115372119467693931463762559748498766540410775079424
1052599132235830049302165227737747090781129258939069605119662354560878748012029885644684086908292942444420519483970025638286418901303230300714082802807490807389380535840308179438902058911320946494923990397355877376
1124721592150840123031562194227404152455344871536319055442865773317966349678908958884929826145247039071457520729423476186410500650998815714261684425163672952019048382424914010323959264719664384955651064457152929335335559060566555523364375289404334925824
1211619148310894857824180097958568721998617225307546483947269775864846125922457987829420626387552743239916249557574345273421200812723858706119388867714044190003158462308987743268219219128754556769517417672969043716622030017691712607256306313531015953731497215009463529699912126963771415920784510976
13546099970612058317732740783210571240950962591140571917288107107573186296098293862183056377562425247549695688432031605439724062025754421207149336809919849514024754412193486549700784652518492833638857753380683723292660549546021427714010561136032121343445343229578372537386497239469170782216646646293140525551530189762671995319084778138053812224
14256666986187667409334369206425788891358044102960677080267979645397319296991771206157830149555422143627981288769976298827679609321487147569940454615893382770261590762252853605038718187735671873757336738248914223947336207483149874735012832025641988716378181005687728396099675895969210085637653343839452156301718734927674663042992666581290686486383686109418989131512359209147653612273582215982104576
Technical details
Node id
n-758efadc25824013 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-1794ed9cb86d5232

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:               47,               27;
  2:             2209,             2538,               729;
  3:           103823,           178929,            102789,             19683;
  4:          4879681,         11212884,           9662166,           3700404,            531441;
  5:        229345007,        658756935,         756869670,         434797470,         124888635,         14348907;
  6:      10779215329,      37153891134,       53359311735,       40870962180,       17609297535,       4046391774,        387420489;
  7:     506623120463,    2037271697181,     3511042712163,     3361636639305,     1931152963005,     665631446823,     127461340881,     10460353203;
  8:   23811286661761,  109430594020008,   220025343295548,   252795075275736,   181528378522470,   83425808001816,   23962732085628,   3933092804328,    282429536481;
  9: 1119130473102767, 5786142658807923, 13295817173430972, 17822052806939388, 15357300823000962, 8822279196192042, 3378745224073548, 831849128115372, 119467693931463, 7625597484987;
%F Formula
The row polynomials satisfy P(n,x) = (47+27*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[(47 + 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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (47 + 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

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.