Real-rooted atlas

A146986: Triangle, read by rows, T(n, k) = binomial(n, k) for n < 2 and binomial(n, k) + 2^(n-1) * binomial(n-2, k-1) otherwise.

A146986 ↗ palindromicγ-positivenonnegative coefficients

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

Recurrence from A146986

\(\left(-3 + n\right)\,P_{n}(x) = \left(-9 - 9\,x + 3\,n + 3\,n x\right)\,P_{n-1}(x) + \left(6 + 12\,x + 6\,x^{2} - 2\,n - 4\,n x - 2\,n x^{2}\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + x\)
\(P_{2}(x) = 1 + 4\,x + x^{2}\)
\(P_{3}(x) = 1 + 7\,x + 7\,x^{2} + x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1112
21416
3177116
41122212148
51215858211160
6138143212143381576
71713416756753417112176

All 15 rows b-files: A146986, A146986 (reversed triangle) ?

Technical details
Node id
n-02236a48f5253afe · revision 0 · JSON
Definition source
oeis_catalog:A146986
Fingerprint
f6-a5de81f788c8c901

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: yes
  • γ-positive (gamma_positive): yes
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

  • A146986 reversed triangle 66 terms, 11 rows · tabl · nonn · catalog b-file?
  • A146986 triangle 66 terms, 11 rows · tabl · nonn · catalog b-file?

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1;
  1: 1,   1;
  2: 1,   4,    1;
  3: 1,   7,    7,    1;
  4: 1,  12,   22,   12,    1;
  5: 1,  21,   58,   58,   21,    1;
  6: 1,  38,  143,  212,  143,   38,    1;
  7: 1,  71,  341,  675,  675,  341,   71,    1;
  8: 1, 136,  796, 1976, 2630, 1976,  796,  136,   1;
  9: 1, 265, 1828, 5460, 9086, 9086, 5460, 1828, 265, 1;
%F Formula
The row polynomials satisfy (n-3)*P(n,x) = (3*n-9+(3*n-9)*x)*P(n-1,x) - (2*n-6+(4*n-12)*x+(2*n-6)*x^2)*P(n-2,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + x, P(2,x) = 1 + 4*x + x^2, P(3,x) = 1 + 7*x + 7*x^2 + x^3.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + x;
P[2, x_] := 1 + 4*x + x^2;
P[3, x_] := 1 + 7*x + 7*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[Together[((3*n - 9 + (3*n - 9)*x)*P[n - 1, x] - (2*n - 6 + (4*n - 12)*x + (2*n - 6)*x^2)*P[n - 2, x])/(n - 3)]];
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; v[3] = 1 + 4*x + x^2; v[4] = 1 + 7*x + 7*x^2 + x^3; for(n = 4, N, my(i = n + 1); v[i] = ((3*n - 9 + (3*n - 9)*x)*v[i - 1] - (2*n - 6 + (4*n - 12)*x + (2*n - 6)*x^2)*v[i - 2])/(n - 3)); 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

A146986 triangle, offset 0, from P0
%e Example
Triangle begins:
  0: 1;
  1: 1,   1;
  2: 1,   4,    1;
  3: 1,   7,    7,    1;
  4: 1,  12,   22,   12,    1;
  5: 1,  21,   58,   58,   21,    1;
  6: 1,  38,  143,  212,  143,   38,    1;
  7: 1,  71,  341,  675,  675,  341,   71,    1;
  8: 1, 136,  796, 1976, 2630, 1976,  796,  136,   1;
  9: 1, 265, 1828, 5460, 9086, 9086, 5460, 1828, 265, 1;
%F Formula
The row polynomials satisfy (n-3)*P(n,x) = (3*n-9+(3*n-9)*x)*P(n-1,x) - (2*n-6+(4*n-12)*x+(2*n-6)*x^2)*P(n-2,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + x, P(2,x) = 1 + 4*x + x^2, P(3,x) = 1 + 7*x + 7*x^2 + x^3.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + x;
P[2, x_] := 1 + 4*x + x^2;
P[3, x_] := 1 + 7*x + 7*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[Together[((3*n - 9 + (3*n - 9)*x)*P[n - 1, x] - (2*n - 6 + (4*n - 12)*x + (2*n - 6)*x^2)*P[n - 2, x])/(n - 3)]];
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] = 1 + x; v[3] = 1 + 4*x + x^2; v[4] = 1 + 7*x + 7*x^2 + x^3; for(n = 4, N, my(i = n + 1); v[i] = ((3*n - 9 + (3*n - 9)*x)*v[i - 1] - (2*n - 6 + (4*n - 12)*x + (2*n - 6)*x^2)*v[i - 2])/(n - 3)); 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

A146986 reversed triangle, offset 0, from P0
%e Example
Triangle begins:
  0: 1;
  1: 1,   1;
  2: 1,   4,    1;
  3: 1,   7,    7,    1;
  4: 1,  12,   22,   12,    1;
  5: 1,  21,   58,   58,   21,    1;
  6: 1,  38,  143,  212,  143,   38,    1;
  7: 1,  71,  341,  675,  675,  341,   71,    1;
  8: 1, 136,  796, 1976, 2630, 1976,  796,  136,   1;
  9: 1, 265, 1828, 5460, 9086, 9086, 5460, 1828, 265, 1;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where (n-3)*P(n,x) = (3*n-9+(3*n-9)*x)*P(n-1,x) - (2*n-6+(4*n-12)*x+(2*n-6)*x^2)*P(n-2,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + x, P(2,x) = 1 + 4*x + x^2, P(3,x) = 1 + 7*x + 7*x^2 + x^3.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + x;
P[2, x_] := 1 + 4*x + x^2;
P[3, x_] := 1 + 7*x + 7*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[Together[((3*n - 9 + (3*n - 9)*x)*P[n - 1, x] - (2*n - 6 + (4*n - 12)*x + (2*n - 6)*x^2)*P[n - 2, x])/(n - 3)]];
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] = 1 + x; v[3] = 1 + 4*x + x^2; v[4] = 1 + 7*x + 7*x^2 + x^3; for(n = 4, N, my(i = n + 1); v[i] = ((3*n - 9 + (3*n - 9)*x)*v[i - 1] - (2*n - 6 + (4*n - 12)*x + (2*n - 6)*x^2)*v[i - 2])/(n - 3)); 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

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 5

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