Real-rooted atlas

A167194: Triangle read by rows. A130713 in the columns.

A167194 ↗ nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence from A167194

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1213
21214
301214
4001214
50001214
600001214
7000001214
80000001214
900000001214
10000000001214
110000000001214
1200000000001214
13000000000001214
140000000000001214

b-files: A113311 (row sums), A167194 ?

Technical details
Node id
n-b442c9393792f928 · revision 1 · JSON
Definition source
oeis_catalog:A167194
Fingerprint
f6-a836c7bd99609ac8

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): yes

Matches ?

  • A167194 triangle 91 terms, 13 rows · tabl · nonn · catalog b-file?

Row sums ? A113311

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1;
  1: 2, 1;
  2: 1, 2, 1;
  3: 0, 1, 2, 1;
  4: 0, 0, 1, 2, 1;
  5: 0, 0, 0, 1, 2, 1;
  6: 0, 0, 0, 0, 1, 2, 1;
  7: 0, 0, 0, 0, 0, 1, 2, 1;
  8: 0, 0, 0, 0, 0, 0, 1, 2, 1;
  9: 0, 0, 0, 0, 0, 0, 0, 1, 2, 1;
%F Formula
The row polynomials satisfy (n-2)*P(n,x) = (n-2)*x*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 2 + x, P(2,x) = 1 + 2*x + x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + x;
P[2, x_] := 1 + 2*x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n - 2)*x*P[n - 1, x])/(n - 2)]];
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 + x; v[3] = 1 + 2*x + x^2; for(n = 3, N, my(i = n + 1); v[i] = ((n - 2)*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 - 1, v[i])))};
T(9)

polytool c48629374597

A167194 triangle, offset 1, from P0
%e Example
Triangle begins:
   1: 1;
   2: 2, 1;
   3: 1, 2, 1;
   4: 0, 1, 2, 1;
   5: 0, 0, 1, 2, 1;
   6: 0, 0, 0, 1, 2, 1;
   7: 0, 0, 0, 0, 1, 2, 1;
   8: 0, 0, 0, 0, 0, 1, 2, 1;
   9: 0, 0, 0, 0, 0, 0, 1, 2, 1;
  10: 0, 0, 0, 0, 0, 0, 0, 1, 2, 1;
%F Formula
The row polynomials satisfy (n-3)*P(n,x) = (n-3)*x*P(n-1,x) for n >= 4, with P(1,x) = 1, P(2,x) = 2 + x, P(3,x) = 1 + 2*x + x^2.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2 + x;
P[3, x_] := 1 + 2*x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n - 3)*x*P[n - 1, x])/(n - 3)]];
Table[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + x; v[3] = 1 + 2*x + x^2; for(n = 4, N, my(i = n); v[i] = ((n - 3)*x*v[i - 1])/(n - 3)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

Used in 5

References

  1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.