Real-rooted atlas

A007318 → (θ+1)² → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n: all roots real, in [−1, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence
\(\left(1 + 3\,x + 3\,x^{2} + x^{3}\right)\,P_{n}(x) = \left(1 + 9\,x + 33\,x^{2} + 63\,x^{3} + 66\,x^{4} + 36\,x^{5} + 8\,x^{6}\right)\,P_{n-3}(x) + \left(-3 - 21\,x - 57\,x^{2} - 75\,x^{3} - 48\,x^{4} - 12\,x^{5}\right)\,P_{n-2}(x) + \left(3 + 15\,x + 27\,x^{2} + 21\,x^{3} + 6\,x^{4}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 5\,x\)
\(P_{2}(x) = 1 + 10\,x + 18\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1156
21101829
31155456126
4120108224160513
51251805608004321998
613027011202400259211207533
71353781960560090727840281627702
8140504313611200241923136022528691299873
914564847042016054432940801013766220816640354294
10150810672033600108864235200337920311040166400394241240029
111559909240528001995845174409292801140480915200433664921604290894
121601188123207920034214410348802230272342144036608002601984110592021299214703201
1316514041601611440055598419219204832256889574411897600112752647188480276889648742449955454
1417016382038416016086486433633609664512207567363331328039463424335462401938227268239361105920168466797

Generated b-files: A081278 (row sums) ?

Computed in your browser ?

Technical details
Node id
n-8c39fb67b0147cfb · revision 1 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-12c97c7d57838f47

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

OEIS matches ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1;
  1: 1,  5;
  2: 1, 10,  18;
  3: 1, 15,  54,   56;
  4: 1, 20, 108,  224,   160;
  5: 1, 25, 180,  560,   800,   432;
  6: 1, 30, 270, 1120,  2400,  2592,  1120;
  7: 1, 35, 378, 1960,  5600,  9072,  7840,   2816;
  8: 1, 40, 504, 3136, 11200, 24192, 31360,  22528,  6912;
  9: 1, 45, 648, 4704, 20160, 54432, 94080, 101376, 62208, 16640;
%F Formula
The row polynomials satisfy (1+3*x+3*x^2+x^3)*P(n,x) = (1+9*x+33*x^2+63*x^3+66*x^4+36*x^5+8*x^6)*P(n-3,x) - (3+21*x+57*x^2+75*x^3+48*x^4+12*x^5)*P(n-2,x) + (3+15*x+27*x^2+21*x^3+6*x^4)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1 + 5*x, P(2,x) = 1 + 10*x + 18*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 5*x;
P[2, x_] := 1 + 10*x + 18*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((1 + 9*x + 33*x^2 + 63*x^3 + 66*x^4 + 36*x^5 + 8*x^6)*P[n - 3, x] - (3 + 21*x + 57*x^2 + 75*x^3 + 48*x^4 + 12*x^5)*P[n - 2, x] + (3 + 15*x + 27*x^2 + 21*x^3 + 6*x^4)*P[n - 1, x])/(1 + 3*x + 3*x^2 + x^3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 5*x; v[3] = 1 + 10*x + 18*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((1 + 9*x + 33*x^2 + 63*x^3 + 66*x^4 + 36*x^5 + 8*x^6)*v[i - 3] - (3 + 21*x + 57*x^2 + 75*x^3 + 48*x^4 + 12*x^5)*v[i - 2] + (3 + 15*x + 27*x^2 + 21*x^3 + 6*x^4)*v[i - 1])/(1 + 3*x + 3*x^2 + x^3)); 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)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-5fe876e73621d007
  • Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
    All coefficients of each polynomial have the same sign (after x -> -x when alternating).
    derived from the cited steps by a stated theorem: e-5fe876e73621d007
  • Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
    Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
    derived from the cited steps by a stated theorem: e-5fe876e73621d007

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

No candidate equalities or related sequences.

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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  4. [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
  5. [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
  6. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.