Real-rooted atlas

A023531 → (1/2)D²+2D+1 → 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
\(\left(1 + 3\,x + 3\,x^{2} + x^{3}\right)\,P_{n}(x) = \left(x^{3} + 3\,x^{4} + 3\,x^{5} + x^{6}\right)\,P_{n-3}(x) + \left(-3\,x^{2} - 9\,x^{3} - 9\,x^{4} - 3\,x^{5}\right)\,P_{n-2}(x) + \left(3\,x + 9\,x^{2} + 9\,x^{3} + 3\,x^{4}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 3\,x\)
\(P_{2}(x) = 1 + 6\,x + 6\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1235
216613
303121025
4006201541
500010302161
6000015422885
700000215636113
8000000287245145
90000000369055181
10000000004511066221
110000000005513278265
1200000000006615691313
130000000000078182105365
1400000000000091210120421

Generated b-files: A001844 (row sums), A099776 (row sums) ?

Computed in your browser ?

Technical details
Node id
n-116049c4b224c370 · revision 1 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-c8cb8dcbb616687c

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: 2, 3;
  2: 1, 6,  6;
  3: 0, 3, 12, 10;
  4: 0, 0,  6, 20, 15;
  5: 0, 0,  0, 10, 30, 21;
  6: 0, 0,  0,  0, 15, 42, 28;
  7: 0, 0,  0,  0,  0, 21, 56, 36;
  8: 0, 0,  0,  0,  0,  0, 28, 72, 45;
  9: 0, 0,  0,  0,  0,  0,  0, 36, 90, 55;
%F Formula
The row polynomials satisfy (1+3*x+3*x^2+x^3)*P(n,x) = (x^3+3*x^4+3*x^5+x^6)*P(n-3,x) - (3*x^2+9*x^3+9*x^4+3*x^5)*P(n-2,x) + (3*x+9*x^2+9*x^3+3*x^4)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 2 + 3*x, P(2,x) = 1 + 6*x + 6*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 3*x;
P[2, x_] := 1 + 6*x + 6*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((x^3 + 3*x^4 + 3*x^5 + x^6)*P[n - 3, x] - (3*x^2 + 9*x^3 + 9*x^4 + 3*x^5)*P[n - 2, x] + (3*x + 9*x^2 + 9*x^3 + 3*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] = 2 + 3*x; v[3] = 1 + 6*x + 6*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((x^3 + 3*x^4 + 3*x^5 + x^6)*v[i - 3] - (3*x^2 + 9*x^3 + 9*x^4 + 3*x^5)*v[i - 2] + (3*x + 9*x^2 + 9*x^3 + 3*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-41edf74927d938cc
  • 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-41edf74927d938cc
  • 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-41edf74927d938cc

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