Real-rooted atlas

basis-map coefficients of A297191, shifted → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 - 12\,x + 9\,x^{2} + 18\,n x - 36\,n x^{2} + 36\,n^{2} x^{2}\right)\,P_{n}(x) = \left(1 + 12\,x + 54\,x^{2} + 108\,x^{3} + 81\,x^{4} + 18\,n x + 144\,n x^{2} + 378\,n x^{3} + 324\,n x^{4} + 36\,n^{2} x^{2} + 216\,n^{2} x^{3} + 324\,n^{2} x^{4}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27x28Pn(1)
011
112481106
2148486172820254288
317212158640303755248835721128512
4196226824192141750489888100018811197445314413309568
5112036455184042525022044967501410167961602391484519683000714492977725696
611445346950401002375692841633006204110854656263063295433026000471565314306110016898135291715470336
711687371157248202702517513496107270163480370176159591732339405366007152073929928533715281730311394374822312107616802536205232128
8119297202419203685500382112642860537681601233920683964567022517352000572165914321114240458241634606227801749928924801291401630005877312307212440502369738734374912
9121612393352512619650074952864663124644445434163223254795278956987460003126478031828118037624321667298352356267739125494432930741565002997429276672190339686245775314543061613985879652914680198217728
10124015390492480981112513562899213845459601085031936066945622770330595668000132006850232442845198572801131381024813024224016116160417122726490005695115625676860274233977805476992106056802657317134051092980917360001537671920841285598145314816
11126418711665280148128752303698322665250973238707025921699388885709790717860004620239758134179949834005765807755927373415543743674536034412624935425062646271882444892822320325819711018517649912081023067096609635715953063672000355202213714271111381840905544166005805331615460174743535616
121288223568743682151765037182499248079037164844348467239085944371125735601232001401303487082463674556648192242869793326524780013318940352211064099603940048028808443207689149628717830799144814803398844481882443457761728419760304557347200163393018308564661024712936331004845817602271524361301435304104448176518460300625102879103987744768
131312263251123200302737505754434408224045830920709504008302569882756150051765000379519694418151970879134348808673921190233003250055495584800103933079350425002838065953462272066080651851000215130735586401734600218497901347343475305814237197040000354018206335223430333031704307576560248178678970756950140988824611315200573684995977031251489194459957016818530201888518411914592791585882112
14133630618141523241461875859753440134575295401657277107201651774429305135948512700009376368920919054778846528224027322851749233951170019978410528043172202191715000137537042360094720378461915146637595898510030179194160183538237131768519032110494905689200004779245785525516305599457067753637808062541027100630751405329377570307714560361421547465529687518763850195458411687004416313859958981673971159904346241923938245462220135254177883056242688
Technical details
Node id
n-f5469f7bf3514e04 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-1b92fdc3be2f687b

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: 1,  24,    81;
  2: 1,  48,   486,   1728,    2025;
  3: 1,  72,  1215,   8640,   30375,    52488,     35721;
  4: 1,  96,  2268,  24192,  141750,   489888,   1000188,    1119744,      531441;
  5: 1, 120,  3645,  51840,  425250,  2204496,   7501410,   16796160,    23914845,    19683000,      7144929;
  6: 1, 144,  5346,  95040, 1002375,  6928416,  33006204,  110854656,   263063295,   433026000,    471565314,    306110016,      89813529;
  7: 1, 168,  7371, 157248, 2027025, 17513496, 107270163,  480370176,  1595917323,  3940536600,   7152073929,   9285337152,    8173031139,    4374822312,    1076168025;
  8: 1, 192,  9720, 241920, 3685500, 38211264, 286053768, 1601233920,  6839645670, 22517352000,  57216591432, 111424045824,  163460622780,  174992892480,  129140163000,   58773123072,   12440502369;
  9: 1, 216, 12393, 352512, 6196500, 74952864, 663124644, 4454341632, 23254795278, 95698746000, 312647803182, 811803762432, 1667298352356, 2677391254944, 3293074156500, 2997429276672, 1903396862457, 753145430616, 139858796529;
%F Formula
The row polynomials satisfy (1+(18*n-12)*x+(36*n^2-36*n+9)*x^2)*P(n,x) = (1+(18*n+12)*x+(36*n^2+144*n+54)*x^2+(216*n^2+378*n+108)*x^3+(324*n^2+324*n+81)*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[((1 + (18*n + 12)*x + (36*n^2 + 144*n + 54)*x^2 + (216*n^2 + 378*n + 108)*x^3 + (324*n^2 + 324*n + 81)*x^4)*P[n - 1, x])/(1 + (18*n - 12)*x + (36*n^2 - 36*n + 9)*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] = ((1 + (18*n + 12)*x + (36*n^2 + 144*n + 54)*x^2 + (216*n^2 + 378*n + 108)*x^3 + (324*n^2 + 324*n + 81)*x^4)*v[i - 1])/(1 + (18*n - 12)*x + (36*n^2 - 36*n + 9)*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.