Real-rooted atlas

A236076: A skewed version of triangular array A122075.

A236076 ↗ nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence from A236076

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1022
20134
300358
40017816
50004151332
6000112302164
700005315834128

All 15 rows b-files: A000079 (row sums), A209599 (reversed triangle), A236076 ?

Technical details
Node id
n-540fd96c9759fa70 · revision 0 · JSON
Definition source
oeis_catalog:A236076
Fingerprint
f6-4899378506777ad9

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): no

Matches ?

  • A209599 reversed triangle 66 terms, 11 rows · tabl · nonn · catalog b-file?
  • A236076 triangle 75 terms, 11 rows · tabl · nonn · catalog b-file?

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

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1;
  1: 0, 2;
  2: 0, 1, 3;
  3: 0, 0, 3, 5;
  4: 0, 0, 1, 7,  8;
  5: 0, 0, 0, 4, 15, 13;
  6: 0, 0, 0, 1, 12, 30, 21;
  7: 0, 0, 0, 0,  5, 31, 58,  34;
  8: 0, 0, 0, 0,  1, 18, 73, 109,  55;
  9: 0, 0, 0, 0,  0,  6, 54, 162, 201, 89;
%F Formula
The row polynomials satisfy P(n,x) = x*P(n-1,x) + (x+x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2*x;
P[n_, x_] := P[n, x] = Expand[x*P[n - 1, x] + (x + 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] = 2*x; for(n = 2, N, my(i = n + 1); v[i] = x*v[i - 1] + (x + 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

A209599 reversed triangle, offset 0, from P0
%e Example
Triangle begins:
  0:  1;
  1:  2,   0;
  2:  3,   1,   0;
  3:  5,   3,   0,  0;
  4:  8,   7,   1,  0, 0;
  5: 13,  15,   4,  0, 0, 0;
  6: 21,  30,  12,  1, 0, 0, 0;
  7: 34,  58,  31,  5, 0, 0, 0, 0;
  8: 55, 109,  73, 18, 1, 0, 0, 0, 0;
  9: 89, 201, 162, 54, 6, 0, 0, 0, 0, 0;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = x*P(n-1,x) + (x+x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2*x;
P[n_, x_] := P[n, x] = Expand[x*P[n - 1, x] + (x + x^2)*P[n - 2, x]];
Table[Reverse[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] = 2*x; for(n = 2, N, my(i = n + 1); v[i] = x*v[i - 1] + (x + x^2)*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

A236076 triangle, offset 0, from P0
%e Example
Triangle begins:
  0: 1;
  1: 0, 2;
  2: 0, 1, 3;
  3: 0, 0, 3, 5;
  4: 0, 0, 1, 7,  8;
  5: 0, 0, 0, 4, 15, 13;
  6: 0, 0, 0, 1, 12, 30, 21;
  7: 0, 0, 0, 0,  5, 31, 58,  34;
  8: 0, 0, 0, 0,  1, 18, 73, 109,  55;
  9: 0, 0, 0, 0,  0,  6, 54, 162, 201, 89;
%F Formula
The row polynomials satisfy P(n,x) = x*P(n-1,x) + (x+x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2*x;
P[n_, x_] := P[n, x] = Expand[x*P[n - 1, x] + (x + x^2)*P[n - 2, x]];
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] = 2*x; for(n = 2, N, my(i = n + 1); v[i] = x*v[i - 1] + (x + x^2)*v[i - 2]); 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

  • 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