Real-rooted atlas

A056242 → reverse → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(-1 - 2\,x - x^{2} + n + 2\,n x + n x^{2}\right)\,P_{n}(x) = \left(-2 - 15\,x - 32\,x^{2} - 27\,x^{3} - 8\,x^{4} + 2\,n + 11\,n x + 20\,n x^{2} + 15\,n x^{3} + 4\,n x^{4}\right)\,P_{n-1}(x) + \left(-2\,x - 10\,x^{2} - 18\,x^{3} - 14\,x^{4} - 4\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2235
34131027
48406534147
516108269292116801
632272914151512363964365
7646562776620177025036135223787
812815367840220483686536614199804616129627

All 15 rows

Technical details
Node id
n-ea9d85984408e1ce · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-cbe7d293337761ab

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:   2,    3;
   3:   4,   13,    10;
   4:   8,   40,    65,     34;
   5:  16,  108,   269,    292,    116;
   6:  32,  272,   914,   1515,   1236,    396;
   7:  64,  656,  2776,   6201,   7702,   5036,    1352;
   8: 128, 1536,  7840,  22048,  36865,  36614,   19980,   4616;
   9: 256, 3520, 21040,  71368, 150169, 200584,  165976,  77728,  15760;
  10: 512, 7936, 54368, 215984, 548082, 920883, 1023944, 726152, 297856, 53808;
%F Formula
The row polynomials satisfy (n-1+(2*n-2)*x+(n-1)*x^2)*P(n,x) = (2*n-2+(11*n-15)*x+(20*n-32)*x^2+(15*n-27)*x^3+(4*n-8)*x^4)*P(n-1,x) - (2*x+10*x^2+18*x^3+14*x^4+4*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 - 2 + (11*n - 15)*x + (20*n - 32)*x^2 + (15*n - 27)*x^3 + (4*n - 8)*x^4)*P[n - 1, x] - (2*x + 10*x^2 + 18*x^3 + 14*x^4 + 4*x^5)*D[P[n - 1, x], x])/(n - 1 + (2*n - 2)*x + (n - 1)*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] = ((2*n - 2 + (11*n - 15)*x + (20*n - 32)*x^2 + (15*n - 27)*x^3 + (4*n - 8)*x^4)*v[i - 1] - (2*x + 10*x^2 + 18*x^3 + 14*x^4 + 4*x^5)*deriv(v[i - 1], x))/(n - 1 + (2*n - 2)*x + (n - 1)*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 ?