Real-rooted atlas

A030528 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(3 + 6\,x + 3\,x^{2} + 3\,n + 6\,n x + 3\,n x^{2}\right)\,P_{n}(x) = \left(23 + 110\,x + 183\,x^{2} + 128\,x^{3} + 32\,x^{4} - 11\,n - 56\,n x - 111\,n x^{2} - 98\,n x^{3} - 32\,n x^{4}\right)\,P_{n-1}(x) + \left(5 + 43\,x + 131\,x^{2} + 185\,x^{3} + 124\,x^{4} + 32\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1257
23182445
3545129116295
48984358245601925
513200120135054952270412575
62139029581170025392286081305682125
7347396771338429947317166016081663040536375

All 15 rows

Technical details
Node id
n-2b38b764c2be7262 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-3024ad5e34af04cb

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 0)
  • 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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:  1;
  1:  2,    5;
  2:  3,   18,    24;
  3:  5,   45,   129,    116;
  4:  8,   98,   435,    824,    560;
  5: 13,  200,  1201,   3505,   4952,    2704;
  6: 21,  390,  2958,  11700,  25392,   28608,   13056;
  7: 34,  739,  6771,  33842,  99473,  171660,  160816,   63040;
  8: 55, 1370, 14719,  88976, 330565,  771920, 1105120,  886016,  304384;
  9: 89, 2499, 30801, 218505, 982236, 2898573, 5609808, 6860256, 4806912, 1469696;
%F Formula
The row polynomials satisfy (3*n+3+(6*n+6)*x+(3*n+3)*x^2)*P(n,x) = -(11*n-23+(56*n-110)*x+(111*n-183)*x^2+(98*n-128)*x^3+(32*n-32)*x^4)*P(n-1,x) + (5+43*x+131*x^2+185*x^3+124*x^4+32*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(11*n - 23 + (56*n - 110)*x + (111*n - 183)*x^2 + (98*n - 128)*x^3 + (32*n - 32)*x^4)*P[n - 1, x] + (5 + 43*x + 131*x^2 + 185*x^3 + 124*x^4 + 32*x^5)*D[P[n - 1, x], x])/(3*n + 3 + (6*n + 6)*x + (3*n + 3)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(11*n - 23 + (56*n - 110)*x + (111*n - 183)*x^2 + (98*n - 128)*x^3 + (32*n - 32)*x^4)*v[i - 1] + (5 + 43*x + 131*x^2 + 185*x^3 + 124*x^4 + 32*x^5)*deriv(v[i - 1], x))/(3*n + 3 + (6*n + 6)*x + (3*n + 3)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

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