Real-rooted atlas

A038314 → basis A013609 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(11 + 24\,x + 12\,n x\right)\,P_{n}(x) = \left(242 + 1056\,x + 864\,x^{2} + 264\,n x + 288\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 484 + 2112\,x + 1728\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0484211217284324
1106486969611404855296249688
2234256204441650181124866048165888013821712
351536325622144018399744026763264018247680047775744743257696
41133799041484246016607191552011775836160120434688006306398208133772083239132965440
5249435788838095647744187014998016453369692160618231398400485592662016206009008128366917713922027499535744
6548758735369578334289925485773275136159586131640322720218152960028488102838272181287927152646457751764992990677827584103724602417408
7120726921779223706377367552155169015496704526634234413056107720638857216014101610904944641196500319207424639317424734208196154209861632264180754022405252474404767232
82655992279142457948922454016042671479261593601655136165298176039497567580979200620470879817564166580751755640832046883277813841920215769630847795205811976588492800697437190619136263746386523350016
958431830141132814023639233871872114738866458951680500678690002698240136548733637099520025025658819308421123185083849730162688283643830773743616017405416888388812807032491672076288001687798001298309121826017371802828813150431719660738560
1012855002631049216336567341612924928302910607451632435214686574906745815040450610821002428416009438248468996318822414014368938812715827214976394264853662924811487575146336616448061885926714271334400222789336171376803844820685861559468032474764516668735488651752364325379313664
1128281005788308275280215216417747107847875675793742443315242003604233293031014414319410534077169664003374173827666183979008572587073785776675225671387479329135793274886570892983704544608256442484376007040040960021239250048337921966086893580782030039285761357826517672583495681227083673851500953632134903999132774064128
12622182127342782054419004835889743160934420214234537272271175681176100918532204868403244103784444957682565120115471726546798296170496220446023407524019961856314104909048197490409472337305839830166623223808272570375620336665231360163542225372201999138816707740960288417366671362091052837215778583347237794177154626229370883155358018475288166401577303165007712586186752
131368800680154120519684479711316868030791680513130569023065345228803234277525963563388108801323113533348730476953600381056697604434377362636880830208582758807319347201295682749823814647939072015901561020564997951979520149913706591185165877248001079378687456533194316185658388629223794432750387202300158120937356441681920623603923051332784619520104126814609684509491200807771652729673770598477114434318544608009486336
1430113614963390651432961051238922358364559114241290156859257992868003840875742837799364855857152038811330311562760657305600121938143233419000756043776284522334211311001764102144506755919931091951193948160699668684904859909887098880753851781715673976982732800633235496641166140665495552411055949735512806562725888202413914642487366868008960731695269713563800620236801832631937130447367045120028433562176084516725063682059817714460668115025923756954460149784104501575680
Technical details
Node id
n-a02b693528ef1fd1 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-7a38cc12383be86e

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:             484,              2112,               1728;
  1:           10648,             69696,             114048,              55296;
  2:          234256,           2044416,            5018112,            4866048,             1658880;
  3:         5153632,          56221440,          183997440,          267632640,           182476800,            47775744;
  4:       113379904,        1484246016,         6071915520,        11775836160,         12043468800,          6306398208,          1337720832;
  5:      2494357888,       38095647744,       187014998016,       453369692160,        618231398400,        485592662016,        206009008128,         36691771392;
  6:     54875873536,      957833428992,      5485773275136,     15958613164032,      27202181529600,      28488102838272,      18128792715264,       6457751764992,        990677827584;
  7:   1207269217792,    23706377367552,    155169015496704,    526634234413056,    1077206388572160,    1410161090494464,    1196500319207424,     639317424734208,     196154209861632,     26418075402240;
  8:  26559922791424,   579489224540160,   4267147926159360,  16551361652981760,   39497567580979200,   62047087981756416,   65807517556408320,   46883277813841920,   21576963084779520,   5811976588492800,    697437190619136;
  9: 584318301411328, 14023639233871872, 114738866458951680, 500678690002698240, 1365487336370995200, 2502565881930842112, 3185083849730162688, 2836438307737436160, 1740541688838881280, 703249167207628800, 168779800129830912, 18260173718028288;
%F Formula
The row polynomials satisfy (11+(12*n+24)*x)*P(n,x) = (242+(264*n+1056)*x+(288*n+864)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 484 + 2112*x + 1728*x^2.
%t Mathematica
P[0, x_] := 484 + 2112*x + 1728*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((242 + (264*n + 1056)*x + (288*n + 864)*x^2)*P[n - 1, x])/(11 + (12*n + 24)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 484 + 2112*x + 1728*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((242 + (264*n + 1056)*x + (288*n + 864)*x^2)*v[i - 1])/(11 + (12*n + 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

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.