Real-rooted atlas

A081577 → 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
\(\left(n + 3\,n x + 3\,n x^{2} + n x^{3}\right)\,P_{n}(x) = \left(6 + 54\,x + 180\,x^{2} + 300\,x^{3} + 270\,x^{4} + 126\,x^{5} + 24\,x^{6} - 9\,n - 77\,n x - 255\,n x^{2} - 429\,n x^{3} - 392\,n x^{4} - 186\,n x^{5} - 36\,n x^{6} + 4\,n^{2} + 28\,n^{2} x + 84\,n^{2} x^{2} + 136\,n^{2} x^{3} + 124\,n^{2} x^{4} + 60\,n^{2} x^{5} + 12\,n^{2} x^{6}\right)\,P_{n-1}(x) + \left(1 + 35\,x + 208\,x^{2} + 550\,x^{3} + 785\,x^{4} + 631\,x^{5} + 270\,x^{6} + 48\,x^{7} - n - 18\,n x - 98\,n x^{2} - 256\,n x^{3} - 369\,n x^{4} - 302\,n x^{5} - 132\,n x^{6} - 24\,n x^{7}\right)\,P_{n-1}'(x) + \left(x + 17\,x^{2} + 82\,x^{3} + 190\,x^{4} + 245\,x^{5} + 181\,x^{6} + 72\,x^{7} + 12\,x^{8}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1123
216613
3110241651
4114588844205
5118108272300120819
612217463211369843283277
7126256123231604400313689613107

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-5397a8ff0da3f463 · revision 1 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-3c392e037ce8dc6c

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,  6,   6;
  3: 1, 10,  24,   16;
  4: 1, 14,  58,   88,    44;
  5: 1, 18, 108,  272,   300,   120;
  6: 1, 22, 174,  632,  1136,   984,   328;
  7: 1, 26, 256, 1232,  3160,  4400,  3136,   896;
  8: 1, 30, 354, 2136,  7236, 14256, 16176,  9792,  2448;
  9: 1, 34, 468, 3408, 14484, 37512, 59808, 57216, 30096, 6688;
%F Formula
The row polynomials satisfy (n+3*n*x+3*n*x^2+n*x^3)*P(n,x) = (4*n^2-9*n+6+(28*n^2-77*n+54)*x+(84*n^2-255*n+180)*x^2+(136*n^2-429*n+300)*x^3+(124*n^2-392*n+270)*x^4+(60*n^2-186*n+126)*x^5+(12*n^2-36*n+24)*x^6)*P(n-1,x) - (n-1+(18*n-35)*x+(98*n-208)*x^2+(256*n-550)*x^3+(369*n-785)*x^4+(302*n-631)*x^5+(132*n-270)*x^6+(24*n-48)*x^7)*d/dx P(n-1,x) + (x+17*x^2+82*x^3+190*x^4+245*x^5+181*x^6+72*x^7+12*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[((4*n^2 - 9*n + 6 + (28*n^2 - 77*n + 54)*x + (84*n^2 - 255*n + 180)*x^2 + (136*n^2 - 429*n + 300)*x^3 + (124*n^2 - 392*n + 270)*x^4 + (60*n^2 - 186*n + 126)*x^5 + (12*n^2 - 36*n + 24)*x^6)*P[n - 1, x] - (n - 1 + (18*n - 35)*x + (98*n - 208)*x^2 + (256*n - 550)*x^3 + (369*n - 785)*x^4 + (302*n - 631)*x^5 + (132*n - 270)*x^6 + (24*n - 48)*x^7)*D[P[n - 1, x], x] + (x + 17*x^2 + 82*x^3 + 190*x^4 + 245*x^5 + 181*x^6 + 72*x^7 + 12*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] = ((4*n^2 - 9*n + 6 + (28*n^2 - 77*n + 54)*x + (84*n^2 - 255*n + 180)*x^2 + (136*n^2 - 429*n + 300)*x^3 + (124*n^2 - 392*n + 270)*x^4 + (60*n^2 - 186*n + 126)*x^5 + (12*n^2 - 36*n + 24)*x^6)*v[i - 1] - (n - 1 + (18*n - 35)*x + (98*n - 208)*x^2 + (256*n - 550)*x^3 + (369*n - 785)*x^4 + (302*n - 631)*x^5 + (132*n - 270)*x^6 + (24*n - 48)*x^7)*deriv(v[i - 1], x) + (x + 17*x^2 + 82*x^3 + 190*x^4 + 245*x^5 + 181*x^6 + 72*x^7 + 12*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-5c2436669acea698
  • 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-5c2436669acea698
  • 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-5c2436669acea698

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