Real-rooted atlas

A120910 → basis A007318 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(9 - 3\,x + 9\,n x + n^{2} x^{2}\right)\,P_{n}(x) = \left(27 + 27\,x + 9\,x^{2} + x^{3} + 27\,n x + 15\,n x^{2} + 2\,n x^{3} + 3\,n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
033
191221
2277227126
38132424348696
424312961458576753648
5729486072904320112510818432
6218717496328052592010125194414790624
765616123613778113608070875204123087192436224

All 15 rows

Technical details
Node id
n-fd33170a2f06ac0d · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-1e916cb7d7a3299c

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 ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:     3;
  1:     9,     12;
  2:    27,     72,      27;
  3:    81,    324,     243,      48;
  4:   243,   1296,    1458,     576,      75;
  5:   729,   4860,    7290,    4320,    1125,     108;
  6:  2187,  17496,   32805,   25920,   10125,    1944,    147;
  7:  6561,  61236,  137781,  136080,   70875,   20412,   3087,   192;
  8: 19683, 209952,  551124,  653184,  425250,  163296,  37044,  4608,  243;
  9: 59049, 708588, 2125764, 2939328, 2296350, 1102248, 333396, 62208, 6561, 300;
%F Formula
The row polynomials satisfy (9+(9*n-3)*x+n^2*x^2)*P(n,x) = (27+(27*n+27)*x+(3*n^2+15*n+9)*x^2+(n^2+2*n+1)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 3.
%t Mathematica
P[0, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[((27 + (27*n + 27)*x + (3*n^2 + 15*n + 9)*x^2 + (n^2 + 2*n + 1)*x^3)*P[n - 1, x])/(9 + (9*n - 3)*x + n^2*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 3; for(n = 1, N, my(i = n + 1); v[i] = ((27 + (27*n + 27)*x + (3*n^2 + 15*n + 9)*x^2 + (n^2 + 2*n + 1)*x^3)*v[i - 1])/(9 + (9*n - 3)*x + n^2*x^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

Properties

Proof chains ? JSON

Same fingerprint ?

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.