Real-rooted atlas

A107131: A Motzkin related triangle.

A107131 ↗ nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel

Recurrence from A107131

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1011
20112
300314
4002619
50001010121
600053015151
700003570211127

All 15 rows b-files: A001006 (row sums), A055151 (reversed triangle), A080159 (reversed triangle), A107131 ?

Technical details
Node id
n-de4284a2d5d7d0a6 · revision 0 · JSON
Definition source
oeis_catalog:A107131
Fingerprint
f6-f68b3fe345bb2270

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 ?

  • A055151 reversed triangle 61 terms, 14 rows · tabf · nonn · catalog b-file?
  • A080159 reversed triangle 82 terms, 12 rows · tabl · nonn b-file?
  • A107131 triangle 78 terms, 12 rows · tabl · nonn · catalog b-file?

Row sums ? A086246, A001006, A168049

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1;
  1: 0, 1;
  2: 0, 1, 1;
  3: 0, 0, 3,  1;
  4: 0, 0, 2,  6,  1;
  5: 0, 0, 0, 10, 10,   1;
  6: 0, 0, 0,  5, 30,  15,   1;
  7: 0, 0, 0,  0, 35,  70,  21,   1;
  8: 0, 0, 0,  0, 14, 140, 140,  28,  1;
  9: 0, 0, 0,  0,  0, 126, 420, 252, 36, 1;
%F Formula
The row polynomials satisfy (n+2)*P(n,x) = (2*n+1)*x*P(n-1,x) + ((4*n-4)*x-(n-1)*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n + 1)*x*P[n - 1, x] + ((4*n - 4)*x - (n - 1)*x^2)*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] = x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n + 1)*x*v[i - 1] + ((4*n - 4)*x - (n - 1)*x^2)*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

A055151 reversed triangle, offset 0, from P0, low zeros dropped
%e Example
Triangle begins:
  0: 1;
  1: 1;
  2: 1,  1;
  3: 1,  3;
  4: 1,  6,   2;
  5: 1, 10,  10;
  6: 1, 15,  30,   5;
  7: 1, 21,  70,  35;
  8: 1, 28, 140, 140,  14;
  9: 1, 36, 252, 420, 126;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, down to the lowest nonzero coefficient, where (n+2)*P(n,x) = (2*n+1)*x*P(n-1,x) + ((4*n-4)*x-(n-1)*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n + 1)*x*P[n - 1, x] + ((4*n - 4)*x - (n - 1)*x^2)*P[n - 2, x])/(n + 2)]];
Table[Reverse[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x]], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n + 1)*x*v[i - 1] + ((4*n - 4)*x - (n - 1)*x^2)*v[i - 2])/(n + 2)); v};
row(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

A080159 reversed triangle, offset 0, from P0
%e Example
Triangle begins:
  0: 1;
  1: 1,  0;
  2: 1,  1,   0;
  3: 1,  3,   0,   0;
  4: 1,  6,   2,   0,   0;
  5: 1, 10,  10,   0,   0, 0;
  6: 1, 15,  30,   5,   0, 0, 0;
  7: 1, 21,  70,  35,   0, 0, 0, 0;
  8: 1, 28, 140, 140,  14, 0, 0, 0, 0;
  9: 1, 36, 252, 420, 126, 0, 0, 0, 0, 0;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where (n+2)*P(n,x) = (2*n+1)*x*P(n-1,x) + ((4*n-4)*x-(n-1)*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n + 1)*x*P[n - 1, x] + ((4*n - 4)*x - (n - 1)*x^2)*P[n - 2, x])/(n + 2)]];
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] = x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n + 1)*x*v[i - 1] + ((4*n - 4)*x - (n - 1)*x^2)*v[i - 2])/(n + 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

A107131 triangle, offset 0, from P0
%e Example
Triangle begins:
  0: 1;
  1: 0, 1;
  2: 0, 1, 1;
  3: 0, 0, 3,  1;
  4: 0, 0, 2,  6,  1;
  5: 0, 0, 0, 10, 10,   1;
  6: 0, 0, 0,  5, 30,  15,   1;
  7: 0, 0, 0,  0, 35,  70,  21,   1;
  8: 0, 0, 0,  0, 14, 140, 140,  28,  1;
  9: 0, 0, 0,  0,  0, 126, 420, 252, 36, 1;
%F Formula
The row polynomials satisfy (n+2)*P(n,x) = (2*n+1)*x*P(n-1,x) + ((4*n-4)*x-(n-1)*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n + 1)*x*P[n - 1, x] + ((4*n - 4)*x - (n - 1)*x^2)*P[n - 2, x])/(n + 2)]];
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] = x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n + 1)*x*v[i - 1] + ((4*n - 4)*x - (n - 1)*x^2)*v[i - 2])/(n + 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

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