Real-rooted atlas

A013622 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(9 - 15\,x + 45\,n x + 25\,n^{2} x^{2}\right)\,P_{n}(x) = \left(27 + 135\,x + 225\,x^{2} + 125\,x^{3} + 135\,n x + 375\,n x^{2} + 250\,n x^{3} + 75\,n^{2} x^{2} + 125\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
132023
29120225354
327540202520004592
481216012150240001562554016
5243810060750180000234375112500595968
6729291602733751080000210937520250007656256283264
7218710206011481755670000147656252126250016078125500000064028672
865613499204592700272160008859375017010000019293750012000000031640625635437056
9196831180980177147001224720004784062501148175000173643750016200000008542968751953125006174015488
1059049393660066430125524880000239203125068890500001302328125016200000000128144531255859375000118164062558955137024
1117714712990780243577125216513000011276718750378897750008595365625013365000000014095898437596679687500389941406257031250000554856087552
125314414251528087687765086605200005074523437519486170000051572193750096228000000012686308593751160156250000701894531250253125000000412597656255158255722496
1315943231381746603108929850337760280002198960156259499507875002873307937500625482000000098953207031251131152343750091246289062504935937500000160913085937523925781250047450798686208
14478296944641044010881254475128963016000923563265625443310367500015084866671875375289200000006926724492187595016796875000958086035156256910312500000033791748046875100488281250001373291015625432520386576384
Technical details
Node id
n-5ca8280eb561c6c3 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-5a59897ed23c9ce5

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:     1;
  1:     3,      20;
  2:     9,     120,      225;
  3:    27,     540,     2025,      2000;
  4:    81,    2160,    12150,     24000,     15625;
  5:   243,    8100,    60750,    180000,    234375,     112500;
  6:   729,   29160,   273375,   1080000,   2109375,    2025000,     765625;
  7:  2187,  102060,  1148175,   5670000,  14765625,   21262500,   16078125,    5000000;
  8:  6561,  349920,  4592700,  27216000,  88593750,  170100000,  192937500,  120000000,  31640625;
  9: 19683, 1180980, 17714700, 122472000, 478406250, 1148175000, 1736437500, 1620000000, 854296875, 195312500;
%F Formula
The row polynomials satisfy (9+(45*n-15)*x+25*n^2*x^2)*P(n,x) = (27+(135*n+135)*x+(75*n^2+375*n+225)*x^2+(125*n^2+250*n+125)*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[((27 + (135*n + 135)*x + (75*n^2 + 375*n + 225)*x^2 + (125*n^2 + 250*n + 125)*x^3)*P[n - 1, x])/(9 + (45*n - 15)*x + 25*n^2*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] = ((27 + (135*n + 135)*x + (75*n^2 + 375*n + 225)*x^2 + (125*n^2 + 250*n + 125)*x^3)*v[i - 1])/(9 + (45*n - 15)*x + 25*n^2*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

Same fingerprint ?

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.