Real-rooted atlas

A030528 → basis A013611 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(4 + 8\,x + 4\,x^{2} + 4\,n + 8\,n x + 4\,n x^{2}\right)\,P_{n}(x) = \left(29 + 148\,x + 254\,x^{2} + 180\,x^{3} + 45\,x^{4} - 13\,n - 68\,n x - 142\,n x^{2} - 132\,n x^{3} - 45\,n x^{4}\right)\,P_{n-1}(x) + \left(5 + 49\,x + 162\,x^{2} + 242\,x^{3} + 169\,x^{4} + 45\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1268
23223560
3555191205456
48120648148012003456
5132451794634610785702526208
621478442721284558357555041125198720
73490610146617542200864581905149502407501506816

All 15 rows

Technical details
Node id
n-348850e45d8d5b47 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-3d3f01dc6dc7b699

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,    6;
  2:  3,   22,    35;
  3:  5,   55,   191,    205;
  4:  8,  120,   648,   1480,    1200;
  5: 13,  245,  1794,   6346,   10785,    7025;
  6: 21,  478,  4427,  21284,   55835,   75550,    41125;
  7: 34,  906, 10146,  61754,  220086,  458190,   514950,   240750;
  8: 55, 1680, 22076, 162720,  734530, 2075600,  3579500,  3440000,  1409375;
  9: 89, 3065, 46228, 400260, 2189470, 7835726, 18322020, 26958900, 22628625, 8250625;
%F Formula
The row polynomials satisfy (4*n+4+(8*n+8)*x+(4*n+4)*x^2)*P(n,x) = -(13*n-29+(68*n-148)*x+(142*n-254)*x^2+(132*n-180)*x^3+(45*n-45)*x^4)*P(n-1,x) + (5+49*x+162*x^2+242*x^3+169*x^4+45*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[(-(13*n - 29 + (68*n - 148)*x + (142*n - 254)*x^2 + (132*n - 180)*x^3 + (45*n - 45)*x^4)*P[n - 1, x] + (5 + 49*x + 162*x^2 + 242*x^3 + 169*x^4 + 45*x^5)*D[P[n - 1, x], x])/(4*n + 4 + (8*n + 8)*x + (4*n + 4)*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] = (-(13*n - 29 + (68*n - 148)*x + (142*n - 254)*x^2 + (132*n - 180)*x^3 + (45*n - 45)*x^4)*v[i - 1] + (5 + 49*x + 162*x^2 + 242*x^3 + 169*x^4 + 45*x^5)*deriv(v[i - 1], x))/(4*n + 4 + (8*n + 8)*x + (4*n + 4)*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