Real-rooted atlas

A143495 → reverse → D

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
10
233
371825
4127481167
5181945253241061
6254101980312412156754
73376056701632416835437443996
842128813650616841188608518215309296015
952204429106189084581490798342412825524882065431
1063308456700499548223597549850705069820193968817714714967095
11754470102960117994872314552376726039965310309100888905743590490112657799
1288627017671525527482050587093189228234064600304957928182774988401715701948617880348622
131028558289575514456852415220314652624110383072821774747282241241002105558817017870352563772927135726092
14117114144564569777768123153030943635264440768328012269641384193888544851599719485059844384607861744922072619959931747199
1513314924696150176896722698195502569255260154167613605760581226412981562694116679939640011156592447033428566092342748233166961566520984007113
Technical details
Node id
n-857373ab960b1489 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-6353b8c2a93c5733

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:  3;
   3:  7,   18;
   4: 12,   74,    81;
   5: 18,  194,   525,    324;
   6: 25,  410,  1980,   3124,    1215;
   7: 33,  760,  5670,  16324,   16835,    4374;
   8: 42, 1288, 13650,  61684,  118860,   85182,   15309;
   9: 52, 2044, 29106, 189084,  581490,  798342,  412825,   52488;
  10: 63, 3084, 56700, 499548, 2235975, 4985070, 5069820, 1939688, 177147;
%F Formula
The row polynomials satisfy n*P(n,x) = -(n+1+(2*n^2-2)*x+(n^3-n)*x^2)*P(n-2,x) + ((n+1)*x^2+(n^2+n)*x^3)*d/dx P(n-2,x) + (2*n+1+2*n^2*x)*P(n-1,x) - n*x^2*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 3.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[(-(n + 1 + (2*n^2 - 2)*x + (n^3 - n)*x^2)*P[n - 2, x] + ((n + 1)*x^2 + (n^2 + n)*x^3)*D[P[n - 2, x], x] + (2*n + 1 + 2*n^2*x)*P[n - 1, x] - n*x^2*D[P[n - 1, x], x])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 3; for(n = 3, N, my(i = n); v[i] = (-(n + 1 + (2*n^2 - 2)*x + (n^3 - n)*x^2)*v[i - 2] + ((n + 1)*x^2 + (n^2 + n)*x^3)*deriv(v[i - 2], x) + (2*n + 1 + 2*n^2*x)*v[i - 1] - n*x^2*deriv(v[i - 1], x))/(n)); 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

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.