Real-rooted atlas

A122848 → basis A007318 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(4 + 13\,x + 13\,x^{2} + 4\,x^{3} - 3\,n - 10\,n x - 11\,n x^{2} - 4\,n x^{3}\right)\,P_{n-1}(x) + \left(2 + 10\,x + 18\,x^{2} + 14\,x^{3} + 4\,x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1123
227615
3421362081
4106817118876513
52622073512109803123483
6767623150686583105292138425839
7232270213356362955852055944293446512202905

All 15 rows

Technical details
Node id
n-73cdad053aa28c8c · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-019e49ff32dabbfb

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:    1,     2;
  2:    2,     7,      6;
  3:    4,    21,     36,     20;
  4:   10,    68,    171,    188,      76;
  5:   26,   220,    735,   1210,     980,     312;
  6:   76,   762,   3150,   6865,    8310,    5292,    1384;
  7:  232,  2702,  13356,  36295,   58520,   55944,   29344,    6512;
  8:  764, 10096,  57876, 187880,  377545,  480648,  378392,  168352,  32400;
  9: 2620, 38808, 253476, 957768, 2306241, 3668490, 3853416, 2576592, 994896, 168992;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = -(3*n-4+(10*n-13)*x+(11*n-13)*x^2+(4*n-4)*x^3)*P(n-1,x) + (2+10*x+18*x^2+14*x^3+4*x^4)*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[(-(3*n - 4 + (10*n - 13)*x + (11*n - 13)*x^2 + (4*n - 4)*x^3)*P[n - 1, x] + (2 + 10*x + 18*x^2 + 14*x^3 + 4*x^4)*D[P[n - 1, x], x])/(1 + x)]];
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] = (-(3*n - 4 + (10*n - 13)*x + (11*n - 13)*x^2 + (4*n - 4)*x^3)*v[i - 1] + (2 + 10*x + 18*x^2 + 14*x^3 + 4*x^4)*deriv(v[i - 1], x))/(1 + x)); 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