Real-rooted atlas

A119462: Triangle read by rows: T(n,k) is the number of circular binary words of length n having k occurrences of 01 (0 <= k <= floor(n/2)).

A119462 ↗ nonnegative coefficients

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

Recurrence from A119462

\(\left(-2 + n\right)\,P_{n}(x) = \left(-4 + 2\,n\right)\,P_{n-1}(x) + \left(2 - 2\,x - n + n x\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2\)
\(P_{2}(x) = 2 + 2\,x\)

Rows ?

nx0x1x2x3Pn(1)
011
122
2224
3268
4212216
52201032
623030264
72427014128

All 15 rows b-files: A000079 (row sums), A119462 ?

Technical details
Node id
n-ced86a7ce48c966e · revision 0 · JSON
Definition source
oeis_catalog:A119462
Fingerprint
f6-4e15cddc85973b47

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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): no

Matches ?

  • A119462 triangle 64 terms, 15 rows · tabf · nonn · catalog b-file?

Row sums ? A220051, A166444, A011782, A000079, A050732, A034008, A118655, A131577

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1;
  1: 2;
  2: 2,  2;
  3: 2,  6;
  4: 2, 12,   2;
  5: 2, 20,  10;
  6: 2, 30,  30,   2;
  7: 2, 42,  70,  14;
  8: 2, 56, 140,  56,  2;
  9: 2, 72, 252, 168, 18;
%F Formula
The row polynomials satisfy (n-2)*P(n,x) = (2*n-4)*P(n-1,x) - (n-2-(n-2)*x)*P(n-2,x) for n >= 3, with P(0,x) = 1, P(1,x) = 2, P(2,x) = 2 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2;
P[2, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 4)*P[n - 1, x] - (n - 2 - (n - 2)*x)*P[n - 2, 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; v[3] = 2 + 2*x; for(n = 3, N, my(i = n + 1); v[i] = ((2*n - 4)*v[i - 1] - (n - 2 - (n - 2)*x)*v[i - 2])/(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

A119462 triangle, offset 0, from P0
%e Example
Triangle begins:
  0: 1;
  1: 2;
  2: 2,  2;
  3: 2,  6;
  4: 2, 12,   2;
  5: 2, 20,  10;
  6: 2, 30,  30,   2;
  7: 2, 42,  70,  14;
  8: 2, 56, 140,  56,  2;
  9: 2, 72, 252, 168, 18;
%F Formula
The row polynomials satisfy (n-2)*P(n,x) = (2*n-4)*P(n-1,x) - (n-2-(n-2)*x)*P(n-2,x) for n >= 3, with P(0,x) = 1, P(1,x) = 2, P(2,x) = 2 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2;
P[2, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 4)*P[n - 1, x] - (n - 2 - (n - 2)*x)*P[n - 2, 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; v[3] = 2 + 2*x; for(n = 3, N, my(i = n + 1); v[i] = ((2*n - 4)*v[i - 1] - (n - 2 - (n - 2)*x)*v[i - 2])/(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

Properties

Proof chains ? JSON

  • Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

Same fingerprint ?

References

  1. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003