Real-rooted atlas

A081580 → 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
\(\left(n + 3\,n x + 3\,n x^{2} + n x^{3}\right)\,P_{n}(x) = \left(12 + 108\,x + 360\,x^{2} + 600\,x^{3} + 540\,x^{4} + 252\,x^{5} + 48\,x^{6} - 18\,n - 155\,n x - 513\,n x^{2} - 861\,n x^{3} - 785\,n x^{4} - 372\,n x^{5} - 72\,n x^{6} + 7\,n^{2} + 52\,n^{2} x + 162\,n^{2} x^{2} + 268\,n^{2} x^{3} + 247\,n^{2} x^{4} + 120\,n^{2} x^{5} + 24\,n^{2} x^{6}\right)\,P_{n-1}(x) + \left(1 + 65\,x + 406\,x^{2} + 1090\,x^{3} + 1565\,x^{4} + 1261\,x^{5} + 540\,x^{6} + 96\,x^{7} - n - 30\,n x - 182\,n x^{2} - 496\,n x^{3} - 729\,n x^{4} - 602\,n x^{5} - 264\,n x^{6} - 48\,n x^{7}\right)\,P_{n-1}'(x) + \left(x + 29\,x^{2} + 154\,x^{3} + 370\,x^{4} + 485\,x^{5} + 361\,x^{6} + 144\,x^{7} + 24\,x^{8}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1123
219919
3116422887
4123124202101451
51302557408553422223
6137435198539653567118911179
71446644280129101947214308408855767

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-f623dd49245e9923 · revision 1 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-b123934b80f387c2

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,  9,    9;
  3: 1, 16,   42,    28;
  4: 1, 23,  124,   202,   101;
  5: 1, 30,  255,   740,   855,    342;
  6: 1, 37,  435,  1985,  3965,   3567,   1189;
  7: 1, 44,  664,  4280, 12910,  19472,  14308,   4088;
  8: 1, 51,  942,  7968, 33570,  75042,  90912,  56484,  14121;
  9: 1, 58, 1269, 13392, 74226, 228132, 402906, 407208, 219069, 48682;
%F Formula
The row polynomials satisfy (n+3*n*x+3*n*x^2+n*x^3)*P(n,x) = (7*n^2-18*n+12+(52*n^2-155*n+108)*x+(162*n^2-513*n+360)*x^2+(268*n^2-861*n+600)*x^3+(247*n^2-785*n+540)*x^4+(120*n^2-372*n+252)*x^5+(24*n^2-72*n+48)*x^6)*P(n-1,x) - (n-1+(30*n-65)*x+(182*n-406)*x^2+(496*n-1090)*x^3+(729*n-1565)*x^4+(602*n-1261)*x^5+(264*n-540)*x^6+(48*n-96)*x^7)*d/dx P(n-1,x) + (x+29*x^2+154*x^3+370*x^4+485*x^5+361*x^6+144*x^7+24*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[((7*n^2 - 18*n + 12 + (52*n^2 - 155*n + 108)*x + (162*n^2 - 513*n + 360)*x^2 + (268*n^2 - 861*n + 600)*x^3 + (247*n^2 - 785*n + 540)*x^4 + (120*n^2 - 372*n + 252)*x^5 + (24*n^2 - 72*n + 48)*x^6)*P[n - 1, x] - (n - 1 + (30*n - 65)*x + (182*n - 406)*x^2 + (496*n - 1090)*x^3 + (729*n - 1565)*x^4 + (602*n - 1261)*x^5 + (264*n - 540)*x^6 + (48*n - 96)*x^7)*D[P[n - 1, x], x] + (x + 29*x^2 + 154*x^3 + 370*x^4 + 485*x^5 + 361*x^6 + 144*x^7 + 24*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] = ((7*n^2 - 18*n + 12 + (52*n^2 - 155*n + 108)*x + (162*n^2 - 513*n + 360)*x^2 + (268*n^2 - 861*n + 600)*x^3 + (247*n^2 - 785*n + 540)*x^4 + (120*n^2 - 372*n + 252)*x^5 + (24*n^2 - 72*n + 48)*x^6)*v[i - 1] - (n - 1 + (30*n - 65)*x + (182*n - 406)*x^2 + (496*n - 1090)*x^3 + (729*n - 1565)*x^4 + (602*n - 1261)*x^5 + (264*n - 540)*x^6 + (48*n - 96)*x^7)*deriv(v[i - 1], x) + (x + 29*x^2 + 154*x^3 + 370*x^4 + 485*x^5 + 361*x^6 + 144*x^7 + 24*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-28cdd807988aaaf4
  • 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-28cdd807988aaaf4
  • 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-28cdd807988aaaf4

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