Real-rooted atlas

A013610 → D+1

nonnegative coefficients only rational roots

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

Recurrence
\(\left(-2 + 3\,x + 3\,n\right)\,P_{n}(x) = \left(1 + 6\,x + 9\,x^{2} + 3\,n + 9\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1437
2724940
3106310827208
413120378432811024
516195900189016202434864
619288175554008505583272922528
7223993024122852835035721204122187102400
82552847882419273710136080142884699846561458752
928675712843092163296398034612360551124236196196832031616
10318401012571280323190979776199017026244002066715787320590498912896
1134102313860111375588060213305453887689382230108256507577955259815617714738797312
12371224184141663201002375423403212798324277136644222003543302600272806388503056531441167772160
134014432386823938216216207818525275212087130494813510411218295348516888014096722262276349321594323721420288
1443168030303334152251351113621608547296751651272483743509776304858567684046376448150803385279178928208847829693087007744

Generated b-files: A081039 (row sums) ?

Computed in your browser ?

Technical details
Node id
n-85507e03b79a9eac · revision 1 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-1f45c55ca90e4ac8

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

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:  4,   3;
  2:  7,  24,    9;
  3: 10,  63,  108,    27;
  4: 13, 120,  378,   432,     81;
  5: 16, 195,  900,  1890,   1620,    243;
  6: 19, 288, 1755,  5400,   8505,   5832,    729;
  7: 22, 399, 3024, 12285,  28350,  35721,  20412,   2187;
  8: 25, 528, 4788, 24192,  73710, 136080, 142884,  69984,   6561;
  9: 28, 675, 7128, 43092, 163296, 398034, 612360, 551124, 236196, 19683;
%F Formula
The row polynomials satisfy (3*n-2+3*x)*P(n,x) = (3*n+1+(9*n+6)*x+9*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[((3*n + 1 + (9*n + 6)*x + 9*x^2)*P[n - 1, x])/(3*n - 2 + 3*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] = ((3*n + 1 + (9*n + 6)*x + 9*x^2)*v[i - 1])/(3*n - 2 + 3*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)

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-27d4809775fa7a40
  • 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-27d4809775fa7a40
  • 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-27d4809775fa7a40
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-27d4809775fa7a40

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. [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
  4. [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
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.