Real-rooted atlas

A118800: Triangle read by rows: T satisfies the matrix products: C*T*C = T^-1 and T*C*T = C^-1, where C is Pascal's triangle.

A118800 ↗ only rational roots

Proved for all n all roots real, in [0, ∞)alternating coefficient signsinterlacing (weak)

Recurrence from A118800

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
21-10
32-310
44-85-10
58-2018-710
616-4856-329-10
732-112160-12050-1110
864-256432-400220-7213-10

All 15 rows b-files: A118800 ?

Technical details
Node id
n-7c1541c019618244 · revision 0 · JSON
Definition source
oeis_catalog:A118800
Fingerprint
f6-4d86f3a82b76e06e

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 -1)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): no
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): yes

Matches ?

Submission text ?

Atlas rows n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:   1;
   2:   1,    -1;
   3:   2,    -3,    1;
   4:   4,    -8,    5,    -1;
   5:   8,   -20,   18,    -7,    1;
   6:  16,   -48,   56,   -32,    9,    -1;
   7:  32,  -112,  160,  -120,   50,   -11,   1;
   8:  64,  -256,  432,  -400,  220,   -72,  13,   -1;
   9: 128,  -576, 1120, -1232,  840,  -364,  98,  -15,  1;
  10: 256, -1280, 2816, -3584, 2912, -1568, 560, -128, 17, -1;
%F Formula
The row polynomials satisfy (n-2)*P(n,x) = (2*n-4-(n-2)*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 - x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1 - x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 4 - (n - 2)*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] = 1; v[2] = 1 - x; for(n = 3, N, my(i = n); v[i] = ((2*n - 4 - (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, v[i])))};
T(10)

polytool c48629374597

A118800 triangle, offset 0, from P1
%e Example
Triangle begins:
  0:   1;
  1:   1,    -1;
  2:   2,    -3,    1;
  3:   4,    -8,    5,    -1;
  4:   8,   -20,   18,    -7,    1;
  5:  16,   -48,   56,   -32,    9,    -1;
  6:  32,  -112,  160,  -120,   50,   -11,   1;
  7:  64,  -256,  432,  -400,  220,   -72,  13,   -1;
  8: 128,  -576, 1120, -1232,  840,  -364,  98,  -15,  1;
  9: 256, -1280, 2816, -3584, 2912, -1568, 560, -128, 17, -1;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (2*n-2-(n-1)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 - x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 - x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 2 - (n - 1)*x)*P[n - 1, x])/(n - 1)]];
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; v[2] = 1 - x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n - 2 - (n - 1)*x)*v[i - 1])/(n - 1)); 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)

polytool c48629374597

Properties

Proof chains ? JSON

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.