Real-rooted atlas

A081578 → 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
\(\left(n + 3\,n x + 3\,n x^{2} + n x^{3}\right)\,P_{n}(x) = \left(8 + 72\,x + 240\,x^{2} + 400\,x^{3} + 360\,x^{4} + 168\,x^{5} + 32\,x^{6} - 12\,n - 103\,n x - 341\,n x^{2} - 573\,n x^{3} - 523\,n x^{4} - 248\,n x^{5} - 48\,n x^{6} + 5\,n^{2} + 36\,n^{2} x + 110\,n^{2} x^{2} + 180\,n^{2} x^{3} + 165\,n^{2} x^{4} + 80\,n^{2} x^{5} + 16\,n^{2} x^{6}\right)\,P_{n-1}(x) + \left(1 + 45\,x + 274\,x^{2} + 730\,x^{3} + 1045\,x^{4} + 841\,x^{5} + 360\,x^{6} + 64\,x^{7} - n - 22\,n x - 126\,n x^{2} - 336\,n x^{3} - 489\,n x^{4} - 402\,n x^{5} - 176\,n x^{6} - 32\,n x^{7}\right)\,P_{n-1}'(x) + \left(x + 21\,x^{2} + 106\,x^{3} + 250\,x^{4} + 325\,x^{5} + 241\,x^{6} + 96\,x^{7} + 16\,x^{8}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1123
217715
3112302063
41177812261279
51221514044551821215
6127249991186316415475319
71323722008551079445740164023247

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-67cce3037dca960d · revision 1 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-0e2849de7a670bd3

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,  7,   7;
  3: 1, 12,  30,   20;
  4: 1, 17,  78,  122,    61;
  5: 1, 22, 151,  404,   455,   182;
  6: 1, 27, 249,  991,  1863,  1641,    547;
  7: 1, 32, 372, 2008,  5510,  7944,   5740,   1640;
  8: 1, 37, 520, 3580, 13246, 27526,  32140,  19684,  4921;
  9: 1, 42, 693, 5832, 27546, 76572, 127554, 125016, 66429, 14762;
%F Formula
The row polynomials satisfy (n+3*n*x+3*n*x^2+n*x^3)*P(n,x) = (5*n^2-12*n+8+(36*n^2-103*n+72)*x+(110*n^2-341*n+240)*x^2+(180*n^2-573*n+400)*x^3+(165*n^2-523*n+360)*x^4+(80*n^2-248*n+168)*x^5+(16*n^2-48*n+32)*x^6)*P(n-1,x) - (n-1+(22*n-45)*x+(126*n-274)*x^2+(336*n-730)*x^3+(489*n-1045)*x^4+(402*n-841)*x^5+(176*n-360)*x^6+(32*n-64)*x^7)*d/dx P(n-1,x) + (x+21*x^2+106*x^3+250*x^4+325*x^5+241*x^6+96*x^7+16*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[((5*n^2 - 12*n + 8 + (36*n^2 - 103*n + 72)*x + (110*n^2 - 341*n + 240)*x^2 + (180*n^2 - 573*n + 400)*x^3 + (165*n^2 - 523*n + 360)*x^4 + (80*n^2 - 248*n + 168)*x^5 + (16*n^2 - 48*n + 32)*x^6)*P[n - 1, x] - (n - 1 + (22*n - 45)*x + (126*n - 274)*x^2 + (336*n - 730)*x^3 + (489*n - 1045)*x^4 + (402*n - 841)*x^5 + (176*n - 360)*x^6 + (32*n - 64)*x^7)*D[P[n - 1, x], x] + (x + 21*x^2 + 106*x^3 + 250*x^4 + 325*x^5 + 241*x^6 + 96*x^7 + 16*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] = ((5*n^2 - 12*n + 8 + (36*n^2 - 103*n + 72)*x + (110*n^2 - 341*n + 240)*x^2 + (180*n^2 - 573*n + 400)*x^3 + (165*n^2 - 523*n + 360)*x^4 + (80*n^2 - 248*n + 168)*x^5 + (16*n^2 - 48*n + 32)*x^6)*v[i - 1] - (n - 1 + (22*n - 45)*x + (126*n - 274)*x^2 + (336*n - 730)*x^3 + (489*n - 1045)*x^4 + (402*n - 841)*x^5 + (176*n - 360)*x^6 + (32*n - 64)*x^7)*deriv(v[i - 1], x) + (x + 21*x^2 + 106*x^3 + 250*x^4 + 325*x^5 + 241*x^6 + 96*x^7 + 16*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-923647fd37394295
  • 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-923647fd37394295
  • 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-923647fd37394295

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