Real-rooted atlas

A081581 → 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
\(\left(n + 3\,n x + 3\,n x^{2} + n x^{3}\right)\,P_{n}(x) = \left(14 + 126\,x + 420\,x^{2} + 700\,x^{3} + 630\,x^{4} + 294\,x^{5} + 56\,x^{6} - 21\,n - 181\,n x - 599\,n x^{2} - 1005\,n x^{3} - 916\,n x^{4} - 434\,n x^{5} - 84\,n x^{6} + 8\,n^{2} + 60\,n^{2} x + 188\,n^{2} x^{2} + 312\,n^{2} x^{3} + 288\,n^{2} x^{4} + 140\,n^{2} x^{5} + 28\,n^{2} x^{6}\right)\,P_{n-1}(x) + \left(1 + 75\,x + 472\,x^{2} + 1270\,x^{3} + 1825\,x^{4} + 1471\,x^{5} + 630\,x^{6} + 112\,x^{7} - n - 34\,n x - 210\,n x^{2} - 576\,n x^{3} - 849\,n x^{4} - 702\,n x^{5} - 308\,n x^{6} - 56\,n x^{7}\right)\,P_{n-1}'(x) + \left(x + 33\,x^{2} + 178\,x^{3} + 430\,x^{4} + 565\,x^{5} + 421\,x^{6} + 168\,x^{7} + 28\,x^{8}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1123
21101021
3118483299
4126150248124549
513431694411004402835
6142546263253764872162415093
71508405824182002788820608588879299

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-ad4b68eed252b1f1 · revision 1 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-f3ca818af1ab544b

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 ?

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, 10,   10;
  3: 1, 18,   48,    32;
  4: 1, 26,  150,   248,    124;
  5: 1, 34,  316,   944,   1100,    440;
  6: 1, 42,  546,  2632,   5376,   4872,   1624;
  7: 1, 50,  840,  5824,  18200,  27888,  20608,   5888;
  8: 1, 58, 1198, 11032,  48916, 112336, 137872,  86080,  21520;
  9: 1, 66, 1620, 18768, 110964, 354312, 639072, 652800, 352656, 78368;
%F Formula
The row polynomials satisfy (n+3*n*x+3*n*x^2+n*x^3)*P(n,x) = (8*n^2-21*n+14+(60*n^2-181*n+126)*x+(188*n^2-599*n+420)*x^2+(312*n^2-1005*n+700)*x^3+(288*n^2-916*n+630)*x^4+(140*n^2-434*n+294)*x^5+(28*n^2-84*n+56)*x^6)*P(n-1,x) - (n-1+(34*n-75)*x+(210*n-472)*x^2+(576*n-1270)*x^3+(849*n-1825)*x^4+(702*n-1471)*x^5+(308*n-630)*x^6+(56*n-112)*x^7)*d/dx P(n-1,x) + (x+33*x^2+178*x^3+430*x^4+565*x^5+421*x^6+168*x^7+28*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[((8*n^2 - 21*n + 14 + (60*n^2 - 181*n + 126)*x + (188*n^2 - 599*n + 420)*x^2 + (312*n^2 - 1005*n + 700)*x^3 + (288*n^2 - 916*n + 630)*x^4 + (140*n^2 - 434*n + 294)*x^5 + (28*n^2 - 84*n + 56)*x^6)*P[n - 1, x] - (n - 1 + (34*n - 75)*x + (210*n - 472)*x^2 + (576*n - 1270)*x^3 + (849*n - 1825)*x^4 + (702*n - 1471)*x^5 + (308*n - 630)*x^6 + (56*n - 112)*x^7)*D[P[n - 1, x], x] + (x + 33*x^2 + 178*x^3 + 430*x^4 + 565*x^5 + 421*x^6 + 168*x^7 + 28*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] = ((8*n^2 - 21*n + 14 + (60*n^2 - 181*n + 126)*x + (188*n^2 - 599*n + 420)*x^2 + (312*n^2 - 1005*n + 700)*x^3 + (288*n^2 - 916*n + 630)*x^4 + (140*n^2 - 434*n + 294)*x^5 + (28*n^2 - 84*n + 56)*x^6)*v[i - 1] - (n - 1 + (34*n - 75)*x + (210*n - 472)*x^2 + (576*n - 1270)*x^3 + (849*n - 1825)*x^4 + (702*n - 1471)*x^5 + (308*n - 630)*x^6 + (56*n - 112)*x^7)*deriv(v[i - 1], x) + (x + 33*x^2 + 178*x^3 + 430*x^4 + 565*x^5 + 421*x^6 + 168*x^7 + 28*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-1a90f5d53428f521
  • 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-1a90f5d53428f521
  • 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-1a90f5d53428f521

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