Real-rooted atlas

A089231 → D

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
122
261218
3127272156
4202407204801460
53060036007200360015030
642126012600504007560030240170142
7562352352802352007056008467202822402107448

All 15 rows

Technical details
Node id
n-01c7db485d232340 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-8fea465cea2922a0

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:  6,   12;
  3: 12,   72,     72;
  4: 20,  240,    720,     480;
  5: 30,  600,   3600,    7200,     3600;
  6: 42, 1260,  12600,   50400,    75600,    30240;
  7: 56, 2352,  35280,  235200,   705600,   846720,    282240;
  8: 72, 4032,  84672,  846720,  4233600, 10160640,  10160640,   2903040;
  9: 90, 6480, 181440, 2540160, 19051200, 76204800, 152409600, 130636800, 32659200;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (n+1+(2*n^2-2)*x)*P(n-1,x) - (n+1)*x^2*d/dx 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)*x)*P[n - 1, x] - (n + 1)*x^2*D[P[n - 1, x], x])/(n - 1)]];
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)*x)*v[i - 1] - (n + 1)*x^2*deriv(v[i - 1], x))/(n - 1)); 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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.