Real-rooted atlas

basis-map coefficients of A026729 → D

palindromic γ-positive nonnegative coefficients only rational roots

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

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

Rows ?

nx0x1x2x3x4x5x6Pn(1)
10
211
3022
40033
500044
6000055
70000066
800000077

Show all 15 stored rows

Generated b-files: A001477 (row sums), A127648 ?

Computed in your browser ?

Technical details
Node id
n-e9cf5b3881297005 · revision 1 · JSON
Definition source
derived:derivative
Fingerprint
f6-edf6954a4cd8adfc

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -2)
  • Palindromic rows: yes
  • γ-positive (gamma_positive): yes
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 0;
   2: 1;
   3: 0, 2;
   4: 0, 0, 3;
   5: 0, 0, 0, 4;
   6: 0, 0, 0, 0, 5;
   7: 0, 0, 0, 0, 0, 6;
   8: 0, 0, 0, 0, 0, 0, 7;
   9: 0, 0, 0, 0, 0, 0, 0, 8;
  10: 0, 0, 0, 0, 0, 0, 0, 0, 9;
%F Formula
The row polynomials satisfy (n-2)*P(n,x) = (n-1)*x*P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n - 1)*x*P[n - 1, x])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = ((n - 1)*x*v[i - 1])/(n - 2)); 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.

A127648: triangle, offset 0, from the row P2
%e Example
Triangle begins:
  0: 1;
  1: 0, 2;
  2: 0, 0, 3;
  3: 0, 0, 0, 4;
  4: 0, 0, 0, 0, 5;
  5: 0, 0, 0, 0, 0, 6;
  6: 0, 0, 0, 0, 0, 0, 7;
  7: 0, 0, 0, 0, 0, 0, 0, 8;
  8: 0, 0, 0, 0, 0, 0, 0, 0, 9;
  9: 0, 0, 0, 0, 0, 0, 0, 0, 0, 10;
%F Formula
The row polynomials satisfy n*P(n,x) = (n+1)*x*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[((n + 1)*x*P[n - 1, x])/(n)]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {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] = ((n + 1)*x*v[i - 1])/(n)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); 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 >= 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-198d139196a8bb12
  • 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-198d139196a8bb12
  • 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-198d139196a8bb12

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 (75)

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.