Real-rooted atlas

A034870 → basis A013610 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(2 - 3\,x + 6\,n x\right)\,P_{n}(x) = \left(8 + 36\,x + 54\,x^{2} + 27\,x^{3} + 24\,n x + 72\,n x^{2} + 54\,n x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27x28Pn(1)
011
14242755
2161926488644052125
36411526480172802430017496510371875
4256614448384193536453600653184571536279936590492265625
51024307203110401658880544320011757312171460801679616010628820393660064953968359375
64096147456182476812165120513216001478062083017710084434186244676680803464208001714782965101833669087332001953125
7163846881281006387280510976415134720149448499239230231047685922816113487454081260971712010403016624619022476825147788126249746167174453557373046875
86553631457285308416049545216030191616001304277811241845579776102478970880194549921280288222105600332896531968297130788864201182304960999959385603443737680073466403847317942571617431640625
92621441415577627072921628877783042030469120010233564364838802264883211403114577922645878929408489977579520072761670558728659240132608820823804236861197514398723512612433600149871463833644785808528483682825624736098929145013427734375
10104857662914560134479872016137584640128596377600740715134976324062871552011110727024640304676967628806770599280640012288637694361618280618057728022279503257856022147671877632017797236330240011390231251353656728690802640211996491580805594351861160929809173600732224724211239776611328125
1141943042768240646539968512871995801607766212608005032505769984249528411095049777439781683230936430559232080205560709120017204092772106243071143833698304457472466894643256845691152588805873087988979200501170175059558434944873534426241958847582206592861530186618640286381225468800676575645170041012562190050472176437100733855438232421875
12167772161207959552312559534084584206499844512578273280324905635676161800518731038727936980528660482846151611449344843304181170176020871778483961856434684973384990727652266718964940811410480551356006414408642532962304015369218701826457613778264422145203210297941575028940863408621735131904316164872917555201244899187112873637262288593854727968278655917281084529420087047060738412025917911529541015625
1367108864523449139214722007040023555211264002539546214400020113206018048012319338686054406033961805414400241829875482624008060995849420800022611093357625344053818139561951232010931809598521344001901746758559334400283806595346227200036327244204317081603980387499730836480371870112431600640029439717234168384001957211118061056000107891262883115712048440975172011136017264603754487440046996274870438400917895993563250011455341999669366863037736488324735927581787109375
142684354562254857830468491306598411871826477056139122966528000120202243080192080635671399628804344452499898368019244566933143552071276173826457600022345080494594457605983310810122813440137740800941368934402738515332325441536047155557380603904000704189656883684966409118705908474279936010223047454483275776098917449906805770240822028669585643520005826128195688248448034877502123848017920174027205845233395207105836760410286080231309790377939000057734923678333574410376913057570309611956936856459616663426981193869663101673126220703125
Technical details
Node id
n-3e9fa36c1d067fb0 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-875f7e3274313d29

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:      1;
  1:      4,       24,        27;
  2:     16,      192,       648,        864,         405;
  3:     64,     1152,      6480,      17280,       24300,        17496,         5103;
  4:    256,     6144,     48384,     193536,      453600,       653184,       571536,        279936,         59049;
  5:   1024,    30720,    311040,    1658880,     5443200,     11757312,     17146080,      16796160,      10628820,       3936600,        649539;
  6:   4096,   147456,   1824768,   12165120,    51321600,    147806208,    301771008,     443418624,     467668080,     346420800,     171478296,      51018336,       6908733;
  7:  16384,   688128,  10063872,   80510976,   415134720,   1494484992,   3923023104,    7685922816,   11348745408,   12609717120,   10403016624,    6190224768,    2514778812,     624974616,      71744535;
  8:  65536,  3145728,  53084160,  495452160,  3019161600,  13042778112,  41845579776,  102478970880,  194549921280,  288222105600,  332896531968,  297130788864,  201182304960,   99995938560,   34437376800,    7346640384,    731794257;
  9: 262144, 14155776, 270729216, 2887778304, 20304691200, 102335643648, 388022648832, 1140311457792, 2645878929408, 4899775795200, 7276167055872, 8659240132608, 8208238042368, 6119751439872, 3512612433600, 1498714638336, 447858085284, 83682825624, 7360989291;
%F Formula
The row polynomials satisfy (2+(6*n-3)*x)*P(n,x) = (8+(24*n+36)*x+(72*n+54)*x^2+(54*n+27)*x^3)*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[((8 + (24*n + 36)*x + (72*n + 54)*x^2 + (54*n + 27)*x^3)*P[n - 1, x])/(2 + (6*n - 3)*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] = ((8 + (24*n + 36)*x + (72*n + 54)*x^2 + (54*n + 27)*x^3)*v[i - 1])/(2 + (6*n - 3)*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. [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.