Real-rooted atlas

A007318 → θ+1 → D+1

nonnegative coefficients

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

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1325
2510318
372421456
494466365160
51170150140556432
6131022853802557871120
71514048384080542010582816

Show all 15 stored rows

Generated b-files: A001793 (row sums) ?

Computed in your browser ?

Technical details
Node id
n-3e1974518079a8ac · revision 1 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-0bc6c5656007bb44

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:  3,   2;
  2:  5,  10,    3;
  3:  7,  24,   21,    4;
  4:  9,  44,   66,   36,    5;
  5: 11,  70,  150,  140,   55,    6;
  6: 13, 102,  285,  380,  255,   78,    7;
  7: 15, 140,  483,  840,  805,  420,  105,   8;
  8: 17, 184,  756, 1624, 2030, 1512,  644, 136,   9;
  9: 19, 234, 1116, 2856, 4410, 4284, 2604, 936, 171, 10;
%F Formula
The row polynomials satisfy (2*n-1+(n^2+1)*x+n*x^2)*P(n,x) = (2*n+1+(n^2+4*n+3)*x+(n^2+3*n+3)*x^2+(n+1)*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[((2*n + 1 + (n^2 + 4*n + 3)*x + (n^2 + 3*n + 3)*x^2 + (n + 1)*x^3)*P[n - 1, x])/(2*n - 1 + (n^2 + 1)*x + n*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] = ((2*n + 1 + (n^2 + 4*n + 3)*x + (n^2 + 3*n + 3)*x^2 + (n + 1)*x^3)*v[i - 1])/(2*n - 1 + (n^2 + 1)*x + n*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)

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-6a0b4c5cd2e9e2b4
  • 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-6a0b4c5cd2e9e2b4
  • 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-6a0b4c5cd2e9e2b4

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.