Real-rooted atlas

A143685 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n: all roots real, in [−1, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence
\(\left(n + 3\,n x + 3\,n x^{2} + n x^{3}\right)\,P_{n}(x) = \left(20 + 180\,x + 600\,x^{2} + 1000\,x^{3} + 900\,x^{4} + 420\,x^{5} + 80\,x^{6} - 30\,n - 259\,n x - 857\,n x^{2} - 1437\,n x^{3} - 1309\,n x^{4} - 620\,n x^{5} - 120\,n x^{6} + 11\,n^{2} + 84\,n^{2} x + 266\,n^{2} x^{2} + 444\,n^{2} x^{3} + 411\,n^{2} x^{4} + 200\,n^{2} x^{5} + 40\,n^{2} x^{6}\right)\,P_{n-1}(x) + \left(1 + 105\,x + 670\,x^{2} + 1810\,x^{3} + 2605\,x^{4} + 2101\,x^{5} + 900\,x^{6} + 160\,x^{7} - n - 46\,n x - 294\,n x^{2} - 816\,n x^{3} - 1209\,n x^{4} - 1002\,n x^{5} - 440\,n x^{6} - 80\,n x^{7}\right)\,P_{n-1}'(x) + \left(x + 45\,x^{2} + 250\,x^{3} + 610\,x^{4} + 805\,x^{5} + 601\,x^{6} + 240\,x^{7} + 40\,x^{8}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1123
21131327
31246644135
4135240410205891
5146535170020158065103
615795152451126510371345731347
716814881237641870663364958814168185895

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-a793cd06bbb8aaa5 · revision 1 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-ea9a5170d08e2562

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

OEIS matches ?

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1;
  1: 1,  2;
  2: 1, 13,   13;
  3: 1, 24,   66,    44;
  4: 1, 35,  240,   410,    205;
  5: 1, 46,  535,  1700,   2015,     806;
  6: 1, 57,  951,  5245,  11265,   10371,    3457;
  7: 1, 68, 1488, 12376,  41870,   66336,   49588,   14168;
  8: 1, 79, 2146, 24424, 122386,  298666,  376984,  237796,   59449;
  9: 1, 90, 2925, 42720, 296010, 1031652, 1948170, 2035080, 1108845, 246410;
%F Formula
The row polynomials satisfy (n+3*n*x+3*n*x^2+n*x^3)*P(n,x) = (11*n^2-30*n+20+(84*n^2-259*n+180)*x+(266*n^2-857*n+600)*x^2+(444*n^2-1437*n+1000)*x^3+(411*n^2-1309*n+900)*x^4+(200*n^2-620*n+420)*x^5+(40*n^2-120*n+80)*x^6)*P(n-1,x) - (n-1+(46*n-105)*x+(294*n-670)*x^2+(816*n-1810)*x^3+(1209*n-2605)*x^4+(1002*n-2101)*x^5+(440*n-900)*x^6+(80*n-160)*x^7)*d/dx P(n-1,x) + (x+45*x^2+250*x^3+610*x^4+805*x^5+601*x^6+240*x^7+40*x^8)*d^2/dx^2 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[((11*n^2 - 30*n + 20 + (84*n^2 - 259*n + 180)*x + (266*n^2 - 857*n + 600)*x^2 + (444*n^2 - 1437*n + 1000)*x^3 + (411*n^2 - 1309*n + 900)*x^4 + (200*n^2 - 620*n + 420)*x^5 + (40*n^2 - 120*n + 80)*x^6)*P[n - 1, x] - (n - 1 + (46*n - 105)*x + (294*n - 670)*x^2 + (816*n - 1810)*x^3 + (1209*n - 2605)*x^4 + (1002*n - 2101)*x^5 + (440*n - 900)*x^6 + (80*n - 160)*x^7)*D[P[n - 1, x], x] + (x + 45*x^2 + 250*x^3 + 610*x^4 + 805*x^5 + 601*x^6 + 240*x^7 + 40*x^8)*D[P[n - 1, x], {x, 2}])/(n + 3*n*x + 3*n*x^2 + n*x^3)]];
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] = ((11*n^2 - 30*n + 20 + (84*n^2 - 259*n + 180)*x + (266*n^2 - 857*n + 600)*x^2 + (444*n^2 - 1437*n + 1000)*x^3 + (411*n^2 - 1309*n + 900)*x^4 + (200*n^2 - 620*n + 420)*x^5 + (40*n^2 - 120*n + 80)*x^6)*v[i - 1] - (n - 1 + (46*n - 105)*x + (294*n - 670)*x^2 + (816*n - 1810)*x^3 + (1209*n - 2605)*x^4 + (1002*n - 2101)*x^5 + (440*n - 900)*x^6 + (80*n - 160)*x^7)*deriv(v[i - 1], x) + (x + 45*x^2 + 250*x^3 + 610*x^4 + 805*x^5 + 601*x^6 + 240*x^7 + 40*x^8)*deriv(deriv(v[i - 1], x), x))/(n + 3*n*x + 3*n*x^2 + n*x^3)); 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)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-54eec0a056ff5a9d
  • Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
    All coefficients of each polynomial have the same sign (after x -> -x when alternating).
    derived from the cited steps by a stated theorem: e-54eec0a056ff5a9d
  • Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
    Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
    derived from the cited steps by a stated theorem: e-54eec0a056ff5a9d

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

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