Real-rooted atlas

basis-map coefficients of A112227 → 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
\(\left(x + n\right)\,P_{n}(x) = \left(x + x^{2} + n x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 2\,x + x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
10213
200314
3000415
40000516
500000617
6000000718
70000000819
8000000009110
900000000010111
10000000000011112
110000000000012113
1200000000000013114
13000000000000014115
140000000000000015116
1500000000000000016117

Computed in your browser ?

Technical details
Node id
n-944a7a7d8a55e0b1 · revision 1 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-0da9c89ef73fcf6f

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 1)
  • 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 ?

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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 0, 2, 1;
   2: 0, 0, 3, 1;
   3: 0, 0, 0, 4, 1;
   4: 0, 0, 0, 0, 5, 1;
   5: 0, 0, 0, 0, 0, 6, 1;
   6: 0, 0, 0, 0, 0, 0, 7, 1;
   7: 0, 0, 0, 0, 0, 0, 0, 8, 1;
   8: 0, 0, 0, 0, 0, 0, 0, 0, 9,  1;
   9: 0, 0, 0, 0, 0, 0, 0, 0, 0, 10,  1;
  10: 0, 0, 0, 0, 0, 0, 0, 0, 0,  0, 11, 1;
%F Formula
The row polynomials satisfy (n+x)*P(n,x) = ((n+1)*x+x^2)*P(n-1,x) for n >= 2, with P(1,x) = 2*x + x^2.
%t Mathematica
P[1, x_] := 2*x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[(((n + 1)*x + x^2)*P[n - 1, x])/(n + x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 2*x + x^2; for(n = 2, N, my(i = n); v[i] = (((n + 1)*x + x^2)*v[i - 1])/(n + x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 1, 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-087ffb2de1c6b7b4
  • Coefficients of one sign. Every nonzero row, n >= 1: 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-087ffb2de1c6b7b4
  • Hadamard kernel (Malo). Rows P_n, n >= 1, 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-087ffb2de1c6b7b4

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