Real-rooted atlas

A245962: Triangle read by rows: T(n,k) is the number of induced subgraphs of the Lucas cube Lambda(n) that are isomorphic to the hypercube Q(k).

A245962 ↗ nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence from A245962

\(\left(-2 + n\right)\,P_{n}(x) = \left(-2 + n\right)\,P_{n-1}(x) + \left(-2 - 2\,x + n + n x\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 3 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
111
2325
3437
478217
51115531
6183015265
72956357127
84710480242257
976189171669511
101233403551703521025
11199605715407110112047
12322106814109323154824097
13521187227302054832169138191
14843326252084396207953263216385

b-files: A245962 ?

Technical details
Node id
n-299efe8218728fe3 · revision 0 · JSON
Definition source
oeis_catalog:A245962
Fingerprint
f6-c568ad932ec199be

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 ?

  • A245962 triangle 64 terms, 15 rows · tabf · nonn · catalog b-file?

Row sums ? A166977, A014551

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:  1;
  1:  1;
  2:  3,   2;
  3:  4,   3;
  4:  7,   8,   2;
  5: 11,  15,   5;
  6: 18,  30,  15,  2;
  7: 29,  56,  35,  7;
  8: 47, 104,  80, 24, 2;
  9: 76, 189, 171, 66, 9;
%F Formula
The row polynomials satisfy (n-2)*P(n,x) = (n-2)*P(n-1,x) + (n-2+(n-2)*x)*P(n-2,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 3 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1;
P[2, x_] := 3 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((n - 2)*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] = 1; v[3] = 3 + 2*x; for(n = 3, N, my(i = n + 1); v[i] = ((n - 2)*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

A245962 triangle, offset 0, from P0
%e Example
Triangle begins:
  0:  1;
  1:  1;
  2:  3,   2;
  3:  4,   3;
  4:  7,   8,   2;
  5: 11,  15,   5;
  6: 18,  30,  15,  2;
  7: 29,  56,  35,  7;
  8: 47, 104,  80, 24, 2;
  9: 76, 189, 171, 66, 9;
%F Formula
The row polynomials satisfy (n-2)*P(n,x) = (n-2)*P(n-1,x) + (n-2+(n-2)*x)*P(n-2,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 3 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1;
P[2, x_] := 3 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((n - 2)*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] = 1; v[3] = 3 + 2*x; for(n = 3, N, my(i = n + 1); v[i] = ((n - 2)*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

Used in 10

2 more

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