Real-rooted atlas

A206306: Riordan array (1, x/(1-3*x+2*x^2)).

A206306 ↗ nonnegative coefficients

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

Recurrence from A206306

\(\left(-2 + n\right)\,P_{n}(x) = \left(-6 - 2\,x + 3\,n + n x\right)\,P_{n-1}(x) + \left(4 - 2\,n\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = x\)
\(P_{2}(x) = 3\,x + x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1011
20314
3076114
4015239148
50317248121164
606320119882151560
701275226994201251811912
802551291222317957651772116528
905113084656267683840126023824122288
100102371811832423276167587266193230827176096
11020471639849029746166587036036125862808387301259808
12040953687912683722632523871015944669930203943915475331887040
1308191819363193326565808107156449763440641254963136552805723613028544
14016383180241786258183642226113052426751153730867976421176146255693067839110340096

b-files: A007070 (row sums), A110441, A204089 (row sums), A206306 ?

Technical details
Node id
n-d7abbe24afd98bff · revision 0 · JSON
Definition source
oeis_catalog:A206306
Fingerprint
f6-37c3559a5d3b5b73

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 ?

  • A206306 triangle 55 terms, 10 rows · tabl · nonn · catalog b-file?
  • A110441 triangle 55 terms, 10 rows · tabl · nonn · catalog b-file?

Row sums ? A204089, A007070

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1;
  1: 0,   1;
  2: 0,   3,    1;
  3: 0,   7,    6,    1;
  4: 0,  15,   23,    9,    1;
  5: 0,  31,   72,   48,   12,    1;
  6: 0,  63,  201,  198,   82,   15,    1;
  7: 0, 127,  522,  699,  420,  125,   18,   1;
  8: 0, 255, 1291, 2223, 1795,  765,  177,  21,  1;
  9: 0, 511, 3084, 6562, 6768, 3840, 1260, 238, 24, 1;
%F Formula
The row polynomials satisfy (n-2)*P(n,x) = (3*n-6+(n-2)*x)*P(n-1,x) - (2*n-4)*P(n-2,x) for n >= 3, with P(0,x) = 1, P(1,x) = x, P(2,x) = 3*x + x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := x;
P[2, x_] := 3*x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[((3*n - 6 + (n - 2)*x)*P[n - 1, x] - (2*n - 4)*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; v[3] = 3*x + x^2; for(n = 3, N, my(i = n + 1); v[i] = ((3*n - 6 + (n - 2)*x)*v[i - 1] - (2*n - 4)*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

A110441 triangle, offset 0, from P1, low zeros dropped
%e Example
Triangle begins:
  0:    1;
  1:    3,    1;
  2:    7,    6,     1;
  3:   15,   23,     9,     1;
  4:   31,   72,    48,    12,     1;
  5:   63,  201,   198,    82,    15,    1;
  6:  127,  522,   699,   420,   125,   18,    1;
  7:  255, 1291,  2223,  1795,   765,  177,   21,   1;
  8:  511, 3084,  6562,  6768,  3840, 1260,  238,  24,  1;
  9: 1023, 7181, 18324, 23276, 16758, 7266, 1932, 308, 27, 1;
%F Formula
Row n lists the coefficients of P(n,x) in ascending powers of x, starting at the lowest nonzero coefficient, where (n-1)*P(n,x) = (3*n-3+(n-1)*x)*P(n-1,x) - (2*n-2)*P(n-2,x) for n >= 2, with P(0,x) = x, P(1,x) = 3*x + x^2.
%t Mathematica
P[0, x_] := x;
P[1, x_] := 3*x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[((3*n - 3 + (n - 1)*x)*P[n - 1, x] - (2*n - 2)*P[n - 2, x])/(n - 1)]];
Table[PadRight[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], n + 1], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = x; v[2] = 3*x + x^2; for(n = 2, N, my(i = n + 1); v[i] = ((3*n - 3 + (n - 1)*x)*v[i - 1] - (2*n - 2)*v[i - 2])/(n - 1)); v};
row(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); 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

A206306 triangle, offset 0, from P0
%e Example
Triangle begins:
  0: 1;
  1: 0,   1;
  2: 0,   3,    1;
  3: 0,   7,    6,    1;
  4: 0,  15,   23,    9,    1;
  5: 0,  31,   72,   48,   12,    1;
  6: 0,  63,  201,  198,   82,   15,    1;
  7: 0, 127,  522,  699,  420,  125,   18,   1;
  8: 0, 255, 1291, 2223, 1795,  765,  177,  21,  1;
  9: 0, 511, 3084, 6562, 6768, 3840, 1260, 238, 24, 1;
%F Formula
The row polynomials satisfy (n-2)*P(n,x) = (3*n-6+(n-2)*x)*P(n-1,x) - (2*n-4)*P(n-2,x) for n >= 3, with P(0,x) = 1, P(1,x) = x, P(2,x) = 3*x + x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := x;
P[2, x_] := 3*x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[((3*n - 6 + (n - 2)*x)*P[n - 1, x] - (2*n - 4)*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; v[3] = 3*x + x^2; for(n = 3, N, my(i = n + 1); v[i] = ((3*n - 6 + (n - 2)*x)*v[i - 1] - (2*n - 4)*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 6

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