Real-rooted atlas

A143494 → reverse → D

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6Pn(1)
10
222
35813
49382471
51411019564383
6202508558441602129
72749027305404332538412360
835868714023604303451235489675242

All 15 rows

Technical details
Node id
n-3c4cd12cd1402f36 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-3aec4e853484b5c1

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 -2)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:  0;
   2:  2;
   3:  5,    8;
   4:  9,   38,    24;
   5: 14,  110,   195,     64;
   6: 20,  250,   855,    844,    160;
   7: 27,  490,  2730,   5404,   3325,     384;
   8: 35,  868,  7140,  23604,  30345,   12354,    896;
   9: 44, 1428, 16254,  80724, 177870,  158010,  44135,   2048;
  10: 54, 2220, 33390, 232428, 783300, 1225230, 781515, 153368, 4608;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(n+(2*n^2-4*n)*x+(n^3-3*n^2+2*n)*x^2)*P(n-2,x) + (n*x^2+(n^2-n)*x^3)*d/dx P(n-2,x) + (2*n-1+(2*n^2-4*n+2)*x)*P(n-1,x) - (n-1)*x^2*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 2.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[(-(n + (2*n^2 - 4*n)*x + (n^3 - 3*n^2 + 2*n)*x^2)*P[n - 2, x] + (n*x^2 + (n^2 - n)*x^3)*D[P[n - 2, x], x] + (2*n - 1 + (2*n^2 - 4*n + 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, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 2; for(n = 3, N, my(i = n); v[i] = (-(n + (2*n^2 - 4*n)*x + (n^3 - 3*n^2 + 2*n)*x^2)*v[i - 2] + (n*x^2 + (n^2 - n)*x^3)*deriv(v[i - 2], x) + (2*n - 1 + (2*n^2 - 4*n + 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, v[i])))};
T(10)

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.