Real-rooted atlas

A007318 → D+1 → (θ+1)²

nonnegative coefficients

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

Recurrence
\(\left(4\,x - 2\,x^{2} + n - 4\,n x + 6\,n x^{2} + 3\,n^{2} x - 2\,n^{2} x^{2} + n^{2} x^{3} + n^{3} x^{2}\right)\,P_{n}(x) = \left(1 + 4\,x + 6\,x^{2} + 4\,x^{3} + x^{4} + n + 3\,n x + 7\,n x^{2} + 7\,n x^{3} + 2\,n x^{4} + 3\,n^{2} x + 4\,n^{2} x^{2} + 2\,n^{2} x^{3} + n^{2} x^{4} + n^{3} x^{2} + n^{3} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1246
2316928
34365416110
456416212825384
56100360480250361232
6714467512801125432493712
7819611342800350022686866410656
892561764537687508064411610248129440
910324259294081890022680164646912145810078848
101140036451536036750544325145030720109352000121205824
11124844950237606600011642413582810560053460165002662144525824
1213576653435200111375228096316932304128200475880002395834561691318912
13146768424503361787504169886726727687686254823575001384243369643941963256320
1415784106476988827527572072013243231757184170270112012006056052096644613754882257929856

Computed in your browser ?

Technical details
Node id
n-f9acf9db26f6bbe8 · revision 1 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-329289f0f6a2a0c6

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 ?

No OEIS matches.

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:  2,   4;
  2:  3,  16,    9;
  3:  4,  36,   54,   16;
  4:  5,  64,  162,  128,    25;
  5:  6, 100,  360,  480,   250,    36;
  6:  7, 144,  675, 1280,  1125,   432,    49;
  7:  8, 196, 1134, 2800,  3500,  2268,   686,   64;
  8:  9, 256, 1764, 5376,  8750,  8064,  4116, 1024,   81;
  9: 10, 324, 2592, 9408, 18900, 22680, 16464, 6912, 1458, 100;
%F Formula
The row polynomials satisfy (n+(3*n^2-4*n+4)*x+(n^3-2*n^2+6*n-2)*x^2+n^2*x^3)*P(n,x) = (n+1+(3*n^2+3*n+4)*x+(n^3+4*n^2+7*n+6)*x^2+(n^3+2*n^2+7*n+4)*x^3+(n^2+2*n+1)*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[((n + 1 + (3*n^2 + 3*n + 4)*x + (n^3 + 4*n^2 + 7*n + 6)*x^2 + (n^3 + 2*n^2 + 7*n + 4)*x^3 + (n^2 + 2*n + 1)*x^4)*P[n - 1, x])/(n + (3*n^2 - 4*n + 4)*x + (n^3 - 2*n^2 + 6*n - 2)*x^2 + n^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; for(n = 1, N, my(i = n + 1); v[i] = ((n + 1 + (3*n^2 + 3*n + 4)*x + (n^3 + 4*n^2 + 7*n + 6)*x^2 + (n^3 + 2*n^2 + 7*n + 4)*x^3 + (n^2 + 2*n + 1)*x^4)*v[i - 1])/(n + (3*n^2 - 4*n + 4)*x + (n^3 - 2*n^2 + 6*n - 2)*x^2 + n^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 (-inf, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-a8b2755a293a1f99
  • 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-a8b2755a293a1f99
  • 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-a8b2755a293a1f99

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.