Real-rooted atlas

A145324 → D

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
122
251217
395272133
4141424624801098
5203101740417636009846
627590499520416401403024096408
735102212075736962443004155842822401028952

All 15 rows

Technical details
Node id
n-f5fda5c622d40162 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-2dd87c9153a19409

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • 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:  0;
  1:  2;
  2:  5,   12;
  3:  9,   52,    72;
  4: 14,  142,   462,    480;
  5: 20,  310,  1740,   4176,    3600;
  6: 27,  590,  4995,  20416,   40140,    30240;
  7: 35, 1022, 12075,  73696,  244300,   415584,   282240;
  8: 44, 1652, 25872, 218596, 1073380,  3054024,  4645872,  2903040;
  9: 54, 2532, 50652, 563556, 3805830, 15934584, 40276152, 55998720, 32659200;
%F Formula
The row polynomials satisfy n*P(n,x) = -(n+1+(2*n^2+2*n)*x+(n^3+n^2)*x^2)*P(n-2,x) + (2*n+1+(2*n^2+2*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[(-(n + 1 + (2*n^2 + 2*n)*x + (n^3 + n^2)*x^2)*P[n - 2, x] + (2*n + 1 + (2*n^2 + 2*n)*x)*P[n - 1, x])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = (-(n + 1 + (2*n^2 + 2*n)*x + (n^3 + n^2)*x^2)*v[i - 2] + (2*n + 1 + (2*n^2 + 2*n)*x)*v[i - 1])/(n)); 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

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.