Real-rooted atlas

A099097 → basis A007318 → D+1

nonnegative coefficients

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

Recurrence

\(\left(39 + 87\,x + 15\,x^{2} - 27\,x^{3} - 42\,n - 126\,n x - 42\,n x^{2} + 27\,n x^{3} + 39\,n^{2} x + 27\,n^{2} x^{2} + 3\,n^{3}\right)\,P_{n}(x) = \left(-26 - 161\,x - 47\,x^{2} + 153\,x^{3} + 81\,x^{4} - 108\,n + 20\,n x + 163\,n x^{2} + 63\,n x^{3} + 33\,n^{2} + n^{2} x - 18\,n^{2} x^{2} - 2\,n^{3}\right)\,P_{n-1}(x) + \left(26\,x + 213\,x^{2} + 421\,x^{3} + 315\,x^{4} + 81\,x^{5} + 13\,n + 191\,n x + 412\,n x^{2} + 315\,n x^{3} + 81\,n x^{4} + 13\,n^{2} + 22\,n^{2} x + 9\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2639
32937975
412626716827588
551615431621675814436
620347884117188163253824332580
77807372427185370903369099153729234596
8293821665933959285075613745021555473207621871663776

All 15 rows

Technical details
Node id
n-3f62e22fe90b7537 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-32536808faf4d3dd

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (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:      6,       3;
   3:     29,      37,       9;
   4:    126,     267,     168,       27;
   5:    516,    1543,    1621,      675,       81;
   6:   2034,    7884,   11718,     8163,     2538,      243;
   7:   7807,   37242,   71853,    70903,    36909,     9153,     729;
   8:  29382,  166593,  395928,   507561,   374502,   155547,   32076,    2187;
   9: 108923,  715957, 2023677,  3197429,  3066346,  1805823,  623781,  110079,   6561;
  10: 398988, 2983929, 9781998, 18388935, 21742818, 16647378, 8155134, 2410803, 371790, 19683;
%F Formula
The row polynomials satisfy (3*n^3-42*n+39+(39*n^2-126*n+87)*x+(27*n^2-42*n+15)*x^2+(27*n-27)*x^3)*P(n,x) = -(2*n^3-33*n^2+108*n+26-(n^2+20*n-161)*x+(18*n^2-163*n+47)*x^2-(63*n+153)*x^3-81*x^4)*P(n-1,x) + (13*n^2+13*n+(22*n^2+191*n+26)*x+(9*n^2+412*n+213)*x^2+(315*n+421)*x^3+(81*n+315)*x^4+81*x^5)*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[(-(2*n^3 - 33*n^2 + 108*n + 26 - (n^2 + 20*n - 161)*x + (18*n^2 - 163*n + 47)*x^2 - (63*n + 153)*x^3 - 81*x^4)*P[n - 1, x] + (13*n^2 + 13*n + (22*n^2 + 191*n + 26)*x + (9*n^2 + 412*n + 213)*x^2 + (315*n + 421)*x^3 + (81*n + 315)*x^4 + 81*x^5)*D[P[n - 1, x], x])/(3*n^3 - 42*n + 39 + (39*n^2 - 126*n + 87)*x + (27*n^2 - 42*n + 15)*x^2 + (27*n - 27)*x^3)]];
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] = (-(2*n^3 - 33*n^2 + 108*n + 26 - (n^2 + 20*n - 161)*x + (18*n^2 - 163*n + 47)*x^2 - (63*n + 153)*x^3 - 81*x^4)*v[i - 1] + (13*n^2 + 13*n + (22*n^2 + 191*n + 26)*x + (9*n^2 + 412*n + 213)*x^2 + (315*n + 421)*x^3 + (81*n + 315)*x^4 + 81*x^5)*deriv(v[i - 1], x))/(3*n^3 - 42*n + 39 + (39*n^2 - 126*n + 87)*x + (27*n^2 - 42*n + 15)*x^2 + (27*n - 27)*x^3)); 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. [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.