Real-rooted atlas

A272867 → x ↦ x+1 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(6 + 30\,x + 55\,x^{2} + 44\,x^{3} + 13\,x^{4}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27x28Pn(1)
011
16181337
2362164804681691369
32161944723614256156789126219750653
412961555281216241056444744522288381264158184285611874161
57776116640784080311040080632801427284817470440146016007975110257049037129369343957
6466568398086905088342921601145599202712113284665565445876245445377951803487957201521730084009964448268092565726409
727993658786565715360034096204813943845444135144608917042212815449512320198692479441941220663214182698348751405636827290035506081779346274851794931877133
8167961640310784452376576315095961615246620928543422960641475821347843115215521285165221970886749633629446928161327365527315946883360021144481504538642884680074006490357864488157307213512479453921
910077696272097792346168857627572576256154077894144641455428096206186020992052320575877121062508887302417396959902912230210258915522456160367564820971791116640141362719767367357001485392285252827558477594816122213214837680210604499373129961739795077
1060466176181398528025798901760231283123200146577568896069805781544962592037158912076844749870080184731752545920363650107772160589397588951040787908566839680867212949451680781610719743360571228189009920333312732088128151642217237940518428247454001252962387456019088098871401378584918494808584372417849
1136279705611972302848188231205888187566077952013293136074240712741119298563002277202721281018382703833088282856084359552065060539407705601248360025670208020064059403194880270478005561878403054230877750624028770130432311840224429231513710081433544893814290473736823759009842979697412757780910942810755120198070839288894272959813861021792160394037177917621779460413
12217678233678364164096134960504832014797766320128115967716153344691304711049216325711110533529612440958642984960392052140941063681031893939791728642287818634408243204296678622698240006857266759317619209309470349179520001074003747819425280104956990079742028886399700597358948859403817610980128033696530343862214415495808118088355256321461995176664155713075353332643077001538069680387106645111992232980851224816582952005840035281
131306069401650936706662495364834140161140982229237769797109563105286427618119217152334937317408911361422607129975848965015109845077263361486562679927756288373976805549344025680375428372830289921482254580117811968023514444585787138560321155159025525926403773179831946755276838038288971847630752327605838745498822082394644006826481924814717620238407916544750772317792905179231216692164766657121030924550801713464260135665515258624471087281176565825451751918660554302875106592253243569224216081305397
14783641640963291294892032665572967055368628678108610568056008034873344576837828269015043293957590232924161540010484871372800600517725363447091219795963069674215424557163793702606970881348663968616409303042822280305995779632645123805488206083624968086461933747996057601110157855777984785408132490381031468543865613717660828936348327681227866906785505995008945229233028362714432621267224246359458768345198095600885841600159975994514467936896606991557217190483521836711350840286975642624312693281439127123622507049790567632452686124775639373763856992899012061295995008299689
Technical details
Node id
n-74cff965a91d71b6 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-d4e45bc63e4b1375

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:        1;
  1:        6,        18,         13;
  2:       36,       216,        480,         468,          169;
  3:      216,      1944,       7236,       14256,        15678,         9126,          2197;
  4:     1296,     15552,      81216,      241056,       444744,       522288,        381264,        158184,          28561;
  5:     7776,    116640,     784080,     3110400,      8063280,     14272848,      17470440,      14601600,        7975110,        2570490,         371293;
  6:    46656,    839808,    6905088,    34292160,    114559920,    271211328,     466556544,     587624544,      537795180,      348795720,      152173008,       40099644,        4826809;
  7:   279936,   5878656,   57153600,   340962048,   1394384544,   4135144608,    9170422128,   15449512320,    19869247944,    19412206632,    14182698348,     7514056368,     2729003550,      608177934,      62748517;
  8:  1679616,  40310784,  452376576,  3150959616,  15246620928,  54342296064,  147582134784,  311521552128,   516522197088,   674963362944,   692816132736,   552731594688,   336002114448,   150453864288,   46800740064,    9035786448,    815730721;
  9: 10077696, 272097792, 3461688576, 27572576256, 154077894144, 641455428096, 2061860209920, 5232057587712, 10625088873024, 17396959902912, 23021025891552, 24561603675648, 20971791116640, 14136271976736, 7357001485392, 2852528275584, 775948161222, 132148376802, 10604499373;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (6+30*x+55*x^2+44*x^3+13*x^4)*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[((6 + 30*x + 55*x^2 + 44*x^3 + 13*x^4)*P[n - 1, x])/(1 + 2*x + x^2)]];
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] = ((6 + 30*x + 55*x^2 + 44*x^3 + 13*x^4)*v[i - 1])/(1 + 2*x + x^2)); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.