Real-rooted atlas

A038259 → basis A013610 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(11 + 15\,n x\right)\,P_{n}(x) = \left(121 + 330\,x + 225\,x^{2} + 165\,n x + 225\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1113041
21216606751456
3133110890222751350047996
4146411597204900505940002531251511536
5161051219615089842501633500013921875455625046154576
6177156128989180148240125359370000459421875300712500797343751378239616
7194871713720278102282897925691787250011791828125115774312506139546875136687500040467966656
8214358881467692104033482502900121754556000259420218750339604650000270140062500120285000000230660156251172644285696
923579476915787689787047353825530020089501740005136520331250840521508750089146220625005954107500000228353554687538443359375033621157396736
102593742460170738430730065111510103753156921702000094169539406250184914731925000245152106718750218317275000000125594455078125422876953125006343154296875955592647499776
1128531167061185593501183308753880802837547748440742750016277877525937503729113760487500593268098259375066040975687500005065643021484375255840556640625076752166992187510379707031250026962915269794816
12313842837672110271220141996011555122659745507003104642270000268584979177968757032043091205000013051898161706250017434817581500000016716621970898437511256984492187500050656430214843750137012132812500001686702392578125756090964319481856
13345227121439311223987066921190150216594576691501001443963844610004267516891383281251256977702552893750266631633874856250041552981902575000004780953883676953125402437195595703125024146231735742187509796367496093750002411984421386718752724673095703125021089799422231760896
1437974983358324114499539100451020192777963040087425140202154938245400065719760127302531252150828513257173750051326589520909828125914165601856650000001227111496810417968751239506562434765625009296299218260742187550288019813281250000185722800446777343754195996567382812500437893890380859375585552078076081831936
Technical details
Node id
n-43af0a9e2d0d62c1 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-00e8c7a05fadfbe1

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, 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:         11,          30;
  2:        121,         660,          675;
  3:       1331,       10890,        22275,         13500;
  4:      14641,      159720,       490050,        594000,        253125;
  5:     161051,     2196150,      8984250,      16335000,      13921875,       4556250;
  6:    1771561,    28989180,    148240125,     359370000,     459421875,     300712500,      79734375;
  7:   19487171,   372027810,   2282897925,    6917872500,   11791828125,   11577431250,    6139546875,    1366875000;
  8:  214358881,  4676921040,  33482502900,  121754556000,  259420218750,  339604650000,  270140062500,  120285000000,   23066015625;
  9: 2357947691, 57876897870, 473538255300, 2008950174000, 5136520331250, 8405215087500, 8914622062500, 5954107500000, 2283535546875, 384433593750;
%F Formula
The row polynomials satisfy (11+15*n*x)*P(n,x) = (121+(165*n+330)*x+(225*n+225)*x^2)*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[((121 + (165*n + 330)*x + (225*n + 225)*x^2)*P[n - 1, x])/(11 + 15*n*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] = ((121 + (165*n + 330)*x + (225*n + 225)*x^2)*v[i - 1])/(11 + 15*n*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. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [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
  4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  5. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  6. [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
  7. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.