Real-rooted atlas

A038320 → basis A013611 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(41 + 24\,x + 24\,n\right)\,P_{n}(x) = \left(1105 + 1968\,x + 576\,x^{2} + 408\,n + 576\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1105 + 1968\,x + 576\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0110519685763649
125721795607084813824189953
25551692469216381888022671363317769442177
31144237766620280148152960152755200680140807962624454947521
42285969771647702288479666016071113420805499187200195880550419110297621433397185
5446545026538404292136138406992192268612968960298676367360184772689920548465541124586471424992771786369
6857607826578573664508803686812045056885804750028812893422510080114691725066245912726077440150436262707211007531417645377745814273
7162535148375318524329053912925955766950402654504672440324783345650155525569958524354564128902102384641824498332467204061779093094426418075402242052125783884097
8304770842597293900843561007202222919486469440740764613560320015927028034641920229600591207464962227983409741824014156235779604480547349499740160010831410914918406340338096537691994323564703041
956649428602616980459502445684565149113500529504019561691480931072048890464494981120084094708022909337610102426013128458248402680288169164804671557807269478401605558532571136002859492481538457615216811431690244093911089596482305
101045292837382806516315035437553667211586168352178576644943148960508323840140844178662703718402816090754910912512040365459850996482048415642670254427996163024964903740899328014948984983262330880462400857380487168074866712243915980836520347436056576181058251434645737089
111916827160485913693261313652634356280254514552825837208321204961508626571970563855656189196492595208788676748552712028161464367192553674506240179914621050155748556816210064139922691850241048654499963511767040466408331477784723456131153697729738178560194653451834181550088764883384653578247964912076029263686593
1234963160456157957616440539259232669998454790069364257185504028505629916493767331841012167667246320455270425910009591400430239744492165897918951873576967028962524257637629952075564140841065414393856605175727890447162408963523479119877399537254414246654488775969734656367230353643266899968050310430627911539097621035720123168587776348763869572760138254017
1363478440612894150233125867377642168647396011592970666618805997120657480832371086226048002565506692484439059865607287607204173507277946881554600575484025814384640253113890358318106411008031630331359159369334784003022565633642616575754240217863262040560978467225611531386210507853031014404273996346632790920396801016945594704431415296001293696787574868148224050485728295604610662415209620105161139079666561
1411478316436554361329932437572119535135368947224166536507296380300032014839002453272071676313606311815990762427770060800197030913982804919797678084664068610671044657886003285280945855123701817671680121494667371992691077283840134956080465746642495078400116066520331876476508962816760540842032503779376496643690043587362512969924608012624727670053782411018240278933648833215473909760033118637761916624594534412116574790945106558976660916783871403612127655425
Technical details
Node id
n-b7de04a38e112919 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-b561c2303414350e

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:            1105,             1968,               576;
  1:           25721,            79560,             70848,              13824;
  2:          555169,          2469216,           3818880,            2267136,             331776;
  3:        11442377,         66620280,         148152960,          152755200,           68014080,            7962624;
  4:       228596977,       1647702288,        4796660160,         7111342080,         5499187200,         1958805504,           191102976;
  5:      4465450265,      38404292136,      138406992192,       268612968960,       298676367360,       184772689920,         54846554112,         4586471424;
  6:     85760782657,     857366450880,     3686812045056,      8858047500288,     12893422510080,     11469172506624,       5912726077440,      1504362627072,       110075314176;
  7:   1625351483753,   18524329053912,    92595576695040,    265450467244032,    478334565015552,    556995852435456,     412890210238464,    182449833246720,     40617790930944,      2641807540224;
  8:  30477084259729,  390084356100720,  2222919486469440,   7407646135603200,  15927028034641920,  22960059120746496,   22279834097418240,  14156235779604480,   5473494997401600,   1083141091491840,    63403380965376;
  9: 566494286026169, 8045950244568456, 51491135005295040, 195616914809310720, 488904644949811200, 840947080229093376, 1010242601312845824, 840268028816916480, 467155780726947840, 160555853257113600, 28594924815384576, 1521681143169024;
%F Formula
The row polynomials satisfy (24*n+41+24*x)*P(n,x) = (408*n+1105+(576*n+1968)*x+576*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1105 + 1968*x + 576*x^2.
%t Mathematica
P[0, x_] := 1105 + 1968*x + 576*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((408*n + 1105 + (576*n + 1968)*x + 576*x^2)*P[n - 1, x])/(24*n + 41 + 24*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1105 + 1968*x + 576*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((408*n + 1105 + (576*n + 1968)*x + 576*x^2)*v[i - 1])/(24*n + 41 + 24*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

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.