Real-rooted atlas

A052553 → D → D+1

nonnegative coefficients

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

Recurrence

\(\left(-51 + 63\,x - 12\,x^{2} + 41\,n - 33\,n x + 4\,n x^{2} - 11\,n^{2} + 4\,n^{2} x + n^{3}\right)\,P_{n}(x) = \left(-30 - 86\,x + 128\,x^{2} - 32\,x^{3} + 28\,n + 67\,n x - 62\,n x^{2} + 8\,n x^{3} - 9\,n^{2} - 20\,n^{2} x + 8\,n^{2} x^{2} + n^{3} + 2\,n^{3} x\right)\,P_{n-1}(x) + \left(-10\,x - 27\,x^{2} + 48\,x^{3} - 16\,x^{4} + 6\,n x + 20\,n x^{2} - 16\,n x^{3} - n^{2} x - 4\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 0\)
\(P_{3}(x) = 1\)

Rows ?

nx0x1x2Pn(1)
10
20
311
422
5527
610616
71718338
826441282

All 15 rows

Technical details
Node id
n-08dd159ee1b67a3f · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-3dfa0014f17e9a3d

Observed on the stored rows ?

  • Degrees: -1, -1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6 (half affine: slope 1, intercept -3)
  • 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:  0;
   3:  1;
   4:  2;
   5:  5,   2;
   6: 10,   6;
   7: 17,  18,   3;
   8: 26,  44,  12;
   9: 37,  90,  42,  4;
  10: 50, 162, 120, 20;
%F Formula
The row polynomials satisfy (n^3-11*n^2+41*n-51+(4*n^2-33*n+63)*x+(4*n-12)*x^2)*P(n,x) = (n^3-9*n^2+28*n-30+(2*n^3-20*n^2+67*n-86)*x+(8*n^2-62*n+128)*x^2+(8*n-32)*x^3)*P(n-1,x) - ((n^2-6*n+10)*x+(4*n^2-20*n+27)*x^2+(16*n-48)*x^3+16*x^4)*d/dx P(n-1,x) for n >= 4, with P(1,x) = 0, P(2,x) = 0, P(3,x) = 1.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 0;
P[3, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^3 - 9*n^2 + 28*n - 30 + (2*n^3 - 20*n^2 + 67*n - 86)*x + (8*n^2 - 62*n + 128)*x^2 + (8*n - 32)*x^3)*P[n - 1, x] - ((n^2 - 6*n + 10)*x + (4*n^2 - 20*n + 27)*x^2 + (16*n - 48)*x^3 + 16*x^4)*D[P[n - 1, x], x])/(n^3 - 11*n^2 + 41*n - 51 + (4*n^2 - 33*n + 63)*x + (4*n - 12)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 0; v[3] = 1; for(n = 4, N, my(i = n); v[i] = ((n^3 - 9*n^2 + 28*n - 30 + (2*n^3 - 20*n^2 + 67*n - 86)*x + (8*n^2 - 62*n + 128)*x^2 + (8*n - 32)*x^3)*v[i - 1] - ((n^2 - 6*n + 10)*x + (4*n^2 - 20*n + 27)*x^2 + (16*n - 48)*x^3 + 16*x^4)*deriv(v[i - 1], x))/(n^3 - 11*n^2 + 41*n - 51 + (4*n^2 - 33*n + 63)*x + (4*n - 12)*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.