Real-rooted atlas

A026729 → basis A013609 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(-2 - 4\,x - 2\,x^{2} + 2\,n + 4\,n x + 2\,n x^{2}\right)\,P_{n}(x) = \left(20 + 98\,x + 178\,x^{2} + 142\,x^{3} + 42\,x^{4} - 9\,n - 44\,n x - 82\,n x^{2} - 68\,n x^{3} - 21\,n x^{4}\right)\,P_{n-1}(x) + \left(5 + 37\,x + 102\,x^{2} + 134\,x^{3} + 85\,x^{4} + 21\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2134
32101224
43235745128
5550184294171704
68100492118814046483840
7131941191384468676426245720992
8213652689108732603136855285399315114688

All 15 rows

Technical details
Node id
n-6224cd706e74abd8 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-a24171d38c8d8aa6

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:  1,    3;
   3:  2,   10,    12;
   4:  3,   23,    57,    45;
   5:  5,   50,   184,   294,    171;
   6:  8,  100,   492,  1188,   1404,    648;
   7: 13,  194,  1191,  3844,   6867,   6426,   2457;
   8: 21,  365,  2689, 10873,  26031,  36855,  28539,   9315;
   9: 34,  674,  5790, 28130,  84466, 160374, 187866, 124038,  35316;
  10: 55, 1225, 12024, 68224, 246446, 587370, 923040, 921672, 530307, 133893;
%F Formula
The row polynomials satisfy (2*n-2+(4*n-4)*x+(2*n-2)*x^2)*P(n,x) = -(9*n-20+(44*n-98)*x+(82*n-178)*x^2+(68*n-142)*x^3+(21*n-42)*x^4)*P(n-1,x) + (5+37*x+102*x^2+134*x^3+85*x^4+21*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[(-(9*n - 20 + (44*n - 98)*x + (82*n - 178)*x^2 + (68*n - 142)*x^3 + (21*n - 42)*x^4)*P[n - 1, x] + (5 + 37*x + 102*x^2 + 134*x^3 + 85*x^4 + 21*x^5)*D[P[n - 1, x], x])/(2*n - 2 + (4*n - 4)*x + (2*n - 2)*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] = (-(9*n - 20 + (44*n - 98)*x + (82*n - 178)*x^2 + (68*n - 142)*x^3 + (21*n - 42)*x^4)*v[i - 1] + (5 + 37*x + 102*x^2 + 134*x^3 + 85*x^4 + 21*x^5)*deriv(v[i - 1], x))/(2*n - 2 + (4*n - 4)*x + (2*n - 2)*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

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