Real-rooted atlas

A038277 → x ↦ x+1 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(18 + 47\,x + 29\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1182947
232410448412209
35832281884541424389103823
4104976676512163490417560087072814879681
518895681522152049047120790203606365529020511149229345007
634012224328784832132427224028447329603437385660221520409259482332110779215329
7612220032690448147233371660448896090882401443701977201395578577967494773844617249876309506623120463
8110199605761420350474248009198507522580741741312519732711792066987771742085396237168112248398218849650024641296123811286661761
91983592903682876209710336185355736888326968002701542416839339862060827130047555542429139680707804820120255726817681039918899682145071459758691119130473102767
1035704672266245752419420672041705040799872017917721232537605051801958618240976681711999526413112856318512160120721534360905607293592700971380261128627565642042070723330020152599132235830049
1164268410079232113897904529305691751089759718404434636005053056014289382682948736032230496495984371251926911021308153659757159508648272048137711826411108025851734128998558083300032193439798122005097657058292472159215084012303
121156831381426176223654067075727361981823538809917441064312641212733440385813332439615872099454103473323202561869368796767093529625815092907736053504259943643862619983201861324857287896176089964034768914981842635310109392459064353814783205469041116191483108948578241
132082296486567116843612543079766835242158791643774607362490491580437796249610031146643430012672029090325265947036748862490328349071412275210067886234017060866561216536253277061521376108887504151341926296070171947119753685835230833128279891771048882792659270079755594102606287129586021895460999706120583177327
1437481336758208102484541237354624942088853346245192667545657054898024574968627225278489541443631933448145291074465170289664196844534299574948666883624439044246141911961651094522637636583897792548793020922763308531844420832668544482207617625899827755109087680992104318750680300492048442585678435665567751628297558232675799463481256666986187667409334369
Technical details
Node id
n-6869ca391fcd2062 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-dcbe8377329c0f7d

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:           18,            29;
  2:          324,          1044,            841;
  3:         5832,         28188,          45414,          24389;
  4:       104976,        676512,        1634904,        1756008,          707281;
  5:      1889568,      15221520,       49047120,       79020360,        63655290,        20511149;
  6:     34012224,     328784832,     1324272240,     2844732960,      3437385660,      2215204092,       594823321;
  7:    612220032,    6904481472,    33371660448,    89609088240,    144370197720,    139557857796,     74947738446,     17249876309;
  8:  11019960576,  142035047424,   800919850752,  2580741741312,   5197327117920,   6698777174208,   5396237168112,   2483982188496,   500246412961;
  9: 198359290368, 2876209710336, 18535573688832, 69680027015424, 168393398620608, 271300475555424, 291396807078048, 201202557268176, 81039918899682, 14507145975869;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (18+47*x+29*x^2)*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[Together[((18 + 47*x + 29*x^2)*P[n - 1, x])/(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] = ((18 + 47*x + 29*x^2)*v[i - 1])/(1 + x)); 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.