Real-rooted atlas

basis-map coefficients of A297191, shifted → D+1

nonnegative coefficients only rational roots

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

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
10213
2000415
300000617
40000000819
500000000010111
60000000000012113
7000000000000014115

Show all 15 stored rows

Generated b-files: A005408 (row sums), A176271 (row sums) ?

Computed in your browser ?

Technical details
Node id
n-5e57eff74a9f8acc · revision 1 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-b43ca45c584c54a9

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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: 0, 2, 1;
  2: 0, 0, 0, 4, 1;
  3: 0, 0, 0, 0, 0, 6, 1;
  4: 0, 0, 0, 0, 0, 0, 0, 8, 1;
  5: 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 1;
  6: 0, 0, 0, 0, 0, 0, 0, 0, 0,  0, 0, 12, 1;
  7: 0, 0, 0, 0, 0, 0, 0, 0, 0,  0, 0,  0, 0, 14, 1;
  8: 0, 0, 0, 0, 0, 0, 0, 0, 0,  0, 0,  0, 0,  0, 0, 16, 1;
  9: 0, 0, 0, 0, 0, 0, 0, 0, 0,  0, 0,  0, 0,  0, 0,  0, 0, 18, 1;
%F Formula
The row polynomials satisfy (2*n-2+x)*P(n,x) = (2*n*x^2+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*x^2 + x^3)*P[n - 1, x])/(2*n - 2 + 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] = ((2*n*x^2 + x^3)*v[i - 1])/(2*n - 2 + 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-c2d9e3ce7eb7c6a5
  • 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-c2d9e3ce7eb7c6a5
  • 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-c2d9e3ce7eb7c6a5

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

Same fingerprint (3)

Equal up to content, sign, reversal and a small index shift.

References

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