Real-rooted atlas

A110552: A triangular array related to A077028 and distributing the values of A007582.

A110552 ↗ nonnegative coefficientsonly rational roots

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

Recurrence from A110552

\(P_{n}(x) = \left(3 + 3\,x\right)\,P_{n-1}(x) + \left(-2 - 4\,x - 2\,x^{2}\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1123
215410
311017836
4119514916136
513613419612932528
6169330650645321642080
711347831940257519267691288256

All 15 rows b-files: A007582 (row sums), A110552 ?

Technical details
Node id
n-cf9f646a76b9c283 · revision 0 · JSON
Definition source
oeis_catalog:A110552
Fingerprint
f6-ef4f1d5c279ca087

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 ?

  • A110552 triangle 66 terms, 11 rows · tabl · nonn · catalog b-file?

Row sums ? A007582

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1;
  1: 1,   2;
  2: 1,   5,    4;
  3: 1,  10,   17,     8;
  4: 1,  19,   51,    49,    16;
  5: 1,  36,  134,   196,   129,    32;
  6: 1,  69,  330,   650,   645,   321,    64;
  7: 1, 134,  783,  1940,  2575,  1926,   769,   128;
  8: 1, 263, 1813,  5411,  8995,  8981,  5383,  1793,  256;
  9: 1, 520, 4124, 14392, 28742, 35896, 28700, 14344, 4097, 512;
%F Formula
The row polynomials satisfy P(n,x) = (3+3*x)*P(n-1,x) - (2+4*x+2*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[(3 + 3*x)*P[n - 1, x] - (2 + 4*x + 2*x^2)*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 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (3 + 3*x)*v[i - 1] - (2 + 4*x + 2*x^2)*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

A110552 triangle, offset 1, from P0
%e Example
Triangle begins:
   1: 1;
   2: 1,   2;
   3: 1,   5,    4;
   4: 1,  10,   17,     8;
   5: 1,  19,   51,    49,    16;
   6: 1,  36,  134,   196,   129,    32;
   7: 1,  69,  330,   650,   645,   321,    64;
   8: 1, 134,  783,  1940,  2575,  1926,   769,   128;
   9: 1, 263, 1813,  5411,  8995,  8981,  5383,  1793,  256;
  10: 1, 520, 4124, 14392, 28742, 35896, 28700, 14344, 4097, 512;
%F Formula
The row polynomials satisfy P(n,x) = (3+3*x)*P(n-1,x) - (2+4*x+2*x^2)*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 2*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[(3 + 3*x)*P[n - 1, x] - (2 + 4*x + 2*x^2)*P[n - 2, x]];
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] = 1 + 2*x; for(n = 3, N, my(i = n); v[i] = (3 + 3*x)*v[i - 1] - (2 + 4*x + 2*x^2)*v[i - 2]); 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

  • 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 12

4 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