Real-rooted atlas

A228955: Table: T(n,k) = n!*binomial(n+1,2*k).

A228955 ↗ nonnegative coefficients

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

Recurrence from A228955

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

Rows ?

nx0x1x2x3x4Pn(1)
011
1112
2268
3636648
424240120384
5120180018001203840
67201512025200504046080
750401411203528001411205040645120

All 15 rows b-files: A000165 (row sums), A228955 ?

Technical details
Node id
n-90f6949024558b8d · revision 0 · JSON
Definition source
oeis_catalog:A228955
Fingerprint
f6-4e15cddc85973b47

Observed on the stored rows ?

  • Degrees: 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7 (half affine: slope 1, intercept 1)
  • 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 ?

  • A228955 triangle 41 terms, 11 rows · tabf · nonn · catalog b-file?

Row sums ? A000165, A052576

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:      1;
  1:      1,        1;
  2:      2,        6;
  3:      6,       36,        6;
  4:     24,      240,      120;
  5:    120,     1800,     1800,      120;
  6:    720,    15120,    25200,     5040;
  7:   5040,   141120,   352800,   141120,     5040;
  8:  40320,  1451520,  5080320,  3386880,   362880;
  9: 362880, 16329600, 76204800, 76204800, 16329600, 362880;
%F Formula
The row polynomials satisfy P(n,x) = 2*n*P(n-1,x) - (n^2-n-(n^2-n)*x)*P(n-2,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[2*n*P[n - 1, x] - (n^2 - n - (n^2 - n)*x)*P[n - 2, x]];
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 + x; for(n = 2, N, my(i = n + 1); v[i] = 2*n*v[i - 1] - (n^2 - n - (n^2 - n)*x)*v[i - 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

A228955 triangle, offset 0, from P0
%e Example
Triangle begins:
  0:      1;
  1:      1,        1;
  2:      2,        6;
  3:      6,       36,        6;
  4:     24,      240,      120;
  5:    120,     1800,     1800,      120;
  6:    720,    15120,    25200,     5040;
  7:   5040,   141120,   352800,   141120,     5040;
  8:  40320,  1451520,  5080320,  3386880,   362880;
  9: 362880, 16329600, 76204800, 76204800, 16329600, 362880;
%F Formula
The row polynomials satisfy P(n,x) = 2*n*P(n-1,x) - (n^2-n-(n^2-n)*x)*P(n-2,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[2*n*P[n - 1, x] - (n^2 - n - (n^2 - n)*x)*P[n - 2, x]];
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 + x; for(n = 2, N, my(i = n + 1); v[i] = 2*n*v[i - 1] - (n^2 - n - (n^2 - n)*x)*v[i - 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