Real-rooted atlas

A028262: Elements in 3-Pascal triangle (by row).

A028262 ↗ palindromicγ-positivenonnegative coefficients

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

Recurrence from A028262

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1112
21315
3144110
41585120
51613136140
6171926197180
7182645452681160

All 15 rows b-files: A020714 (row sums), A028262, A028262 (reversed triangle), A084215 (row sums) ?

Technical details
Node id
n-347cb2d3d9dd489a · revision 0 · JSON
Definition source
oeis_catalog:A028262
Fingerprint
f6-d8916025f625474b

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 ?

  • A028262 reversed triangle 78 terms, 12 rows · tabl · nonn · catalog b-file?
  • A028262 triangle 78 terms, 12 rows · tabl · nonn · catalog b-file?

Row sums ? A323831, A020714, A257113, A287798, A084215, A146523

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1;
  1: 1,  1;
  2: 1,  3,  1;
  3: 1,  4,  4,   1;
  4: 1,  5,  8,   5,   1;
  5: 1,  6, 13,  13,   6,   1;
  6: 1,  7, 19,  26,  19,   7,   1;
  7: 1,  8, 26,  45,  45,  26,   8,  1;
  8: 1,  9, 34,  71,  90,  71,  34,  9,  1;
  9: 1, 10, 43, 105, 161, 161, 105, 43, 10, 1;
%F Formula
The row polynomials satisfy (n-2)*P(n,x) = (n-2+(n-2)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1 + x, P(2,x) = 1 + 3*x + x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + x;
P[2, x_] := 1 + 3*x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n - 2 + (n - 2)*x)*P[n - 1, 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] = 1 + x; v[3] = 1 + 3*x + x^2; for(n = 3, N, my(i = n + 1); v[i] = ((n - 2 + (n - 2)*x)*v[i - 1])/(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

A028262 triangle, offset 0, from P0
%e Example
Triangle begins:
  0: 1;
  1: 1,  1;
  2: 1,  3,  1;
  3: 1,  4,  4,   1;
  4: 1,  5,  8,   5,   1;
  5: 1,  6, 13,  13,   6,   1;
  6: 1,  7, 19,  26,  19,   7,   1;
  7: 1,  8, 26,  45,  45,  26,   8,  1;
  8: 1,  9, 34,  71,  90,  71,  34,  9,  1;
  9: 1, 10, 43, 105, 161, 161, 105, 43, 10, 1;
%F Formula
The row polynomials satisfy (n-2)*P(n,x) = (n-2+(n-2)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1 + x, P(2,x) = 1 + 3*x + x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + x;
P[2, x_] := 1 + 3*x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n - 2 + (n - 2)*x)*P[n - 1, 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] = 1 + x; v[3] = 1 + 3*x + x^2; for(n = 3, N, my(i = n + 1); v[i] = ((n - 2 + (n - 2)*x)*v[i - 1])/(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

A028262 reversed triangle, offset 0, from P0
%e Example
Triangle begins:
  0: 1;
  1: 1,  1;
  2: 1,  3,  1;
  3: 1,  4,  4,   1;
  4: 1,  5,  8,   5,   1;
  5: 1,  6, 13,  13,   6,   1;
  6: 1,  7, 19,  26,  19,   7,   1;
  7: 1,  8, 26,  45,  45,  26,   8,  1;
  8: 1,  9, 34,  71,  90,  71,  34,  9,  1;
  9: 1, 10, 43, 105, 161, 161, 105, 43, 10, 1;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where (n-2)*P(n,x) = (n-2+(n-2)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1 + x, P(2,x) = 1 + 3*x + x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + x;
P[2, x_] := 1 + 3*x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n - 2 + (n - 2)*x)*P[n - 1, 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] = 1 + x; v[3] = 1 + 3*x + x^2; for(n = 3, N, my(i = n + 1); v[i] = ((n - 2 + (n - 2)*x)*v[i - 1])/(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

Properties

Proof chains ? JSON

Used in 5

References

  1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.