Real-rooted atlas

A052553 → basis A013610 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-33 + 297\,x - 108\,x^{2} + 123\,n - 405\,n x + 108\,n x^{2} - 117\,n^{2} + 108\,n^{2} x + 27\,n^{3}\right)\,P_{n}(x) = \left(-75 + 765\,x + 1728\,x^{2} - 1296\,x^{3} + 450\,n + 9\,n x - 2106\,n x^{2} + 648\,n x^{3} - 378\,n^{2} - 540\,n^{2} x + 648\,n^{2} x^{2} + 81\,n^{3} + 162\,n^{3} x\right)\,P_{n-1}(x) + \left(-10 + 81\,x + 477\,x^{2} - 1296\,x^{4} + 60\,n + 144\,n x - 540\,n x^{2} - 1296\,n x^{3} - 45\,n^{2} - 243\,n^{2} x - 324\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
211
3538
49615
52033962
6388427149
77322216227484
81355105221081275
924711311593702813754
1044523884239270040510177
117944899106029315283524328688
1214049792250022797212555145878183
1324651918856592775984819510935729215702
14430136972123768201312160461544325103586349
157465702512633584968004876202296354082421871598140
Technical details
Node id
n-af136264cba93a0b · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-0eb52ff90756e31e

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:   1;
   2:   1;
   3:   5,    3;
   4:   9,    6;
   5:  20,   33,    9;
   6:  38,   84,   27;
   7:  73,  222,  162,   27;
   8: 135,  510,  522,  108;
   9: 247, 1131, 1593,  702,  81;
  10: 445, 2388, 4239, 2700, 405;
%F Formula
The row polynomials satisfy (27*n^3-117*n^2+123*n-33+(108*n^2-405*n+297)*x+(108*n-108)*x^2)*P(n,x) = (81*n^3-378*n^2+450*n-75+(162*n^3-540*n^2+9*n+765)*x+(648*n^2-2106*n+1728)*x^2+(648*n-1296)*x^3)*P(n-1,x) - (45*n^2-60*n+10+(243*n^2-144*n-81)*x+(324*n^2+540*n-477)*x^2+1296*n*x^3+1296*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((81*n^3 - 378*n^2 + 450*n - 75 + (162*n^3 - 540*n^2 + 9*n + 765)*x + (648*n^2 - 2106*n + 1728)*x^2 + (648*n - 1296)*x^3)*P[n - 1, x] - (45*n^2 - 60*n + 10 + (243*n^2 - 144*n - 81)*x + (324*n^2 + 540*n - 477)*x^2 + 1296*n*x^3 + 1296*x^4)*D[P[n - 1, x], x])/(27*n^3 - 117*n^2 + 123*n - 33 + (108*n^2 - 405*n + 297)*x + (108*n - 108)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((81*n^3 - 378*n^2 + 450*n - 75 + (162*n^3 - 540*n^2 + 9*n + 765)*x + (648*n^2 - 2106*n + 1728)*x^2 + (648*n - 1296)*x^3)*v[i - 1] - (45*n^2 - 60*n + 10 + (243*n^2 - 144*n - 81)*x + (324*n^2 + 540*n - 477)*x^2 + 1296*n*x^3 + 1296*x^4)*deriv(v[i - 1], x))/(27*n^3 - 117*n^2 + 123*n - 33 + (108*n^2 - 405*n + 297)*x + (108*n - 108)*x^2)); 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. [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
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.