Real-rooted atlas

A099040 → basis A013611 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(2\,n + 4\,n x + 2\,n x^{2}\right)\,P_{n}(x) = \left(11 + 68\,x + 138\,x^{2} + 116\,x^{3} + 35\,x^{4} - 7\,n - 40\,n x - 94\,n x^{2} - 96\,n x^{3} - 35\,n x^{4}\right)\,P_{n-1}(x) + \left(3 + 31\,x + 110\,x^{2} + 174\,x^{3} + 127\,x^{4} + 35\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
121012
2652110168
31620888012002304
4447524744130401310031680
512025522142488560179800143000435456
63288336873204818881475320237160015610005985792
7896264963326722296704940608022825600303520001704000082280448

All 15 rows

Technical details
Node id
n-67a08196a38dcfaa · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-7f5e4f5a0de6aac8

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,     10;
  2:    6,     52,     110;
  3:   16,    208,     880,     1200;
  4:   44,    752,    4744,    13040,     13100;
  5:  120,   2552,   21424,    88560,    179800,     143000;
  6:  328,   8336,   87320,   481888,   1475320,    2371600,    1561000;
  7:  896,  26496,  332672,  2296704,   9406080,   22825600,   30352000,   17040000;
  8: 2448,  82560, 1208256, 10015104,  51385696,  166977920,  335294400,  380048000,  186010000;
  9: 6688, 253344, 4234368, 40963200, 252621760, 1029304000, 2769040000, 4739600000, 4680500000, 2030500000;
%F Formula
The row polynomials satisfy (2*n+4*n*x+2*n*x^2)*P(n,x) = -(7*n-11+(40*n-68)*x+(94*n-138)*x^2+(96*n-116)*x^3+(35*n-35)*x^4)*P(n-1,x) + (3+31*x+110*x^2+174*x^3+127*x^4+35*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[(-(7*n - 11 + (40*n - 68)*x + (94*n - 138)*x^2 + (96*n - 116)*x^3 + (35*n - 35)*x^4)*P[n - 1, x] + (3 + 31*x + 110*x^2 + 174*x^3 + 127*x^4 + 35*x^5)*D[P[n - 1, x], x])/(2*n + 4*n*x + 2*n*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] = (-(7*n - 11 + (40*n - 68)*x + (94*n - 138)*x^2 + (96*n - 116)*x^3 + (35*n - 35)*x^4)*v[i - 1] + (3 + 31*x + 110*x^2 + 174*x^3 + 127*x^4 + 35*x^5)*deriv(v[i - 1], x))/(2*n + 4*n*x + 2*n*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