Real-rooted atlas

A038336 → x ↦ x/(x+1) → x ↦ x+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(34 + 22\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1156 + 1496\,x + 484\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0115614964843136
139304762964936810648175616
213363363458752335702414481282342569834496
345435424146996960190231360123090880398235205153632550731776
415448044165997475968970179936083701798404061999040105134092811337990430840979456
5525233501442378998800644618056495364980257004803222519238401251095704322698441715224943578881727094849536
6178579390489692441096253442093518944563227092598106112219131308211201134326771916836698807326726784653455365487587353696717311574016
760716992766464353587193169408915166852909056138172250341171213410836062525448677599805163523743278347325441038051978670081679201730201612072692177925416169448144896
8206437775405977613357738408622080388945912486348806711223587999744075994737687644160590076786751119363181786595226624011764589091594240285464294134272041047153404928026559922791424303305489096114176
97018884363803238449957941648246579216162863474432716803137497027389880320406029027073984512036781453040819773442379976373229514752109998908006406144035587882002072576076758176867215360993341112399257658431830141132816985107389382393856
1023864206836931010561852985472044054937665944482975685484544142233198575007907840207074803807732101120214383326295063822336161838393379607003136897591089332274135043629963964211402752010439112053941288960202641586929448550423840186697582182412855002631049216951166013805414055936
11811383032455654359046825163155362269020162649769225022998560768628670737701534952652810169673698113065410560118446787778022761840641021893855339804219801666122543580810861281283208888144362880032768115352188196051243008029855860474272086425652686812601656623104568191116292375347228281005788308275253265296773103187132416
12275870231034922482073624990597399634154258432105107512592578942910464272042973769027852238848484076468030181913542656626451899803764829290496608026843927183510781952449633296349513856712704254571792786121815932928109815283162640783343616355284739643837828464648359640932796184199168135229485677585332633613461758755234738995262218212734278205442982856619293778479415296
13937958785518736439050249103717624152441908428804123448570939635452641280115618263851836837201510402244354533594479780970496031949046889992006293815296344548544892070656109772802866412264228150836543488018547373474417446589399040933429906882446658420736036239043443671458503393281065854218931513485393920229890125651895065477120343274848258485844377603173128849448188477440136880068015412051968167039970680451594847256576
14318905987076370389277081633016149250259522654568448160225430185082977588346880483817965656917226443243520101744072189616416737329152015800255916505137658032291841874344084212864369237164032173258696860012672786628608012612213962603863680791347207254083847772156888298291203285673272226212237640990721159649390197486672108584963126505708865772890488832062247172484205433113804808630910470499072658636807446275700038415627059230113614963390651432969354238358105289311446368256
Technical details
Node id
n-0ae86819df3c1ba8 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-94bd4dafe00b9a6d

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:              1156,               1496,                 484;
  1:             39304,              76296,               49368,               10648;
  2:           1336336,            3458752,             3357024,             1448128,              234256;
  3:          45435424,          146996960,           190231360,           123090880,            39823520,             5153632;
  4:        1544804416,         5997475968,          9701799360,          8370179840,          4061999040,          1051340928,           113379904;
  5:       52523350144,       237899880064,        461805649536,        498025700480,        322251923840,        125109570432,         26984417152,          2494357888;
  6:     1785793904896,      9244109625344,      20935189445632,      27092598106112,      21913130821120,      11343267719168,       3669880732672,        678465345536,        54875873536;
  7:    60716992766464,    353587193169408,     915166852909056,    1381722503411712,    1341083606252544,     867759980516352,     374327834732544,     103805197867008,     16792017302016,     1207269217792;
  8:  2064377754059776,  13357738408622080,   38894591248634880,   67112235879997440,   75994737687644160,   59007678675111936,   31817865952266240,   11764589091594240,   2854642941342720,   410471534049280,   26559922791424;
  9: 70188843638032384, 499579416482465792, 1616286347443271680, 3137497027389880320, 4060290270739845120, 3678145304081977344, 2379976373229514752, 1099989080064061440, 355878820020725760, 76758176867215360, 9933411123992576, 584318301411328;
%F Formula
The row polynomials satisfy P(n,x) = (34+22*x)*P(n-1,x) for n >= 1, with P(0,x) = 1156 + 1496*x + 484*x^2.
%t Mathematica
P[0, x_] := 1156 + 1496*x + 484*x^2;
P[n_, x_] := P[n, x] = Expand[(34 + 22*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] = 1156 + 1496*x + 484*x^2; for(n = 1, N, my(i = n + 1); v[i] = (34 + 22*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.