Real-rooted atlas

A038320 → basis A013610 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(17 + 36\,x + 18\,n x\right)\,P_{n}(x) = \left(289 + 1224\,x + 972\,x^{2} + 306\,n x + 324\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 289 + 1224\,x + 972\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
028912249722485
14913312124957223328109025
283521707472168544815863045248804587625
314198571503378047754360674179204461480011337408187578125
42413756930668911212177361802292209280227535480011564156162380855687510628125
5410338673608266738828982121084681932260809025574040068806729152283321825924897760256295960765625
6697575744111817753782465692807790418548557493763068695173600311923838822419265884162566660953948169917964518411516734140625
711858787649722601454108841435857084561647298821609088939020723121601193108683495689825600922905650956297703424151744857131521983592903680443619451953125
82015993900449426916355389203051196304693401148685667649280266055871551120040565695238853124175880392234880288752353652736012898312856179203372107936256003927513949286416945362314453125
9342718963076337983335845778046339707877529620268505274813019207107492568579920012642975016109222415617792666958451213499172533265408080399483470183680315292092039936007344451085165568771220920950784642741533369140625
10582622237229761148054592048974561293300407016042486086119562428435201812410604987879600368452414755183052853100495067658734085507662393572286464410037365697936768021439862258715648007491340106868879361573290678739599361503880795854028824233838500634765625
1199045780329059372726672070235281482598358090459503528134503242329668417924450474930025793240010178497957611931833616764584871360828902420286556482991255142418123651563848805145611845523897940395520055186205453934077952173848620000725729283323576558837403648291521508119396352909010316667236328125
121683778265594009294991907328584592248515341021274468199722910161061314643951361059213033346138791120269164723767959975155249874639992298465984646897429204217026748416718904845366002604108856384693754196282675203283579224509077638144137919905200575745228839550561050165103411269382118932416331776562220051373121536033944270003428955078125
132862423051509815793909240263420765016601010861234038379930220618409225529361839664024554483954842308339600686370045608297936645761413114799781789869564802198555558844177276057602618867650976152343539202396349484553342013696001674625404499629595453448792393956536703758336033617976892640337899520884622016388308230144014336611310014599168001079462498636393349121262561262322662353515625
14486611918756668684811648755677669653896768196395896898885243585601293902379569126310681605565683029764256556976001697205930958700352433152384367225540646844521625666445234667290691009740808904150013318917968033280931152942569298611036160075916351670649874993889284783062312355966844534784228602242869954297716736080205729485873279533056019497791381619854868480029361379962909899096064206447202864210228019246797557608385894775390625
Technical details
Node id
n-f4bde9de69502f54 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-4460f4582fcec7b1

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:            289,            1224,              972;
  1:           4913,           31212,            49572,             23328;
  2:          83521,          707472,          1685448,           1586304,            524880;
  3:        1419857,        15033780,         47754360,          67417920,          44614800,           11337408;
  4:       24137569,       306689112,       1217736180,        2292209280,        2275354800,         1156415616,          238085568;
  5:      410338673,      6082667388,      28982121084,       68193226080,       90255740400,        68806729152,        28332182592,         4897760256;
  6:     6975757441,    118177537824,     656928077904,     1854855749376,     3068695173600,      3119238388224,      1926588416256,       666095394816,       99179645184;
  7:   118587876497,   2260145410884,   14358570845616,    47298821609088,    93902072312160,    119310868349568,     98256009229056,     50956297703424,    15174485713152,     1983592903680;
  8:  2015993900449,  42691635538920,  305119630469340,  1148685667649280,  2660558715511200,   4056569523885312,   4175880392234880,   2887523536527360,  1289831285617920,   337210793625600,   39275139492864;
  9: 34271896307633, 798333584577804, 6339707877529620, 26850527481301920, 71074925685799200, 126429750161092224, 156177926669584512, 134991725332654080, 80399483470183680, 31529209203993600, 7344451085165568, 771220920950784;
%F Formula
The row polynomials satisfy (17+(18*n+36)*x)*P(n,x) = (289+(306*n+1224)*x+(324*n+972)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 289 + 1224*x + 972*x^2.
%t Mathematica
P[0, x_] := 289 + 1224*x + 972*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((289 + (306*n + 1224)*x + (324*n + 972)*x^2)*P[n - 1, x])/(17 + (18*n + 36)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 289 + 1224*x + 972*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((289 + (306*n + 1224)*x + (324*n + 972)*x^2)*v[i - 1])/(17 + (18*n + 36)*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.