A062190 → x ↦ x/(x+1)

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.

Triangle begins:
  0: 1;
  1: 1,   7;
  2: 1,  16,   36;
  3: 1,  27,  135,   165;
  4: 1,  40,  330,   880,    715;
  5: 1,  55,  660,  2860,   5005,    3003;
  6: 1,  72, 1170,  7280,  20475,   26208,   12376;
  7: 1,  91, 1911, 15925,  63700,  129948,  129948,   50388;
  8: 1, 112, 2940, 31360, 166600,  479808,  759696,  620160,  203490;
  9: 1, 135, 4320, 57120, 385560, 1465128, 3255840, 4186080, 2877930, 817190;

The row polynomials satisfy (n^2+5*n+(2*n^2+10*n)*x+(n^2+5*n)*x^2)*P(n,x) = (n^2+5*n+(4*n^2+25*n+25)*x+(5*n^2+35*n+50)*x^2+(2*n^2+15*n+25)*x^3)*P(n-1,x) + ((2*n+5)*x+(6*n+15)*x^2+(6*n+15)*x^3+(2*n+5)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + 5*n + (4*n^2 + 25*n + 25)*x + (5*n^2 + 35*n + 50)*x^2 + (2*n^2 + 15*n + 25)*x^3)*P[n - 1, x] + ((2*n + 5)*x + (6*n + 15)*x^2 + (6*n + 15)*x^3 + (2*n + 5)*x^4)*D[P[n - 1, x], x])/(n^2 + 5*n + (2*n^2 + 10*n)*x + (n^2 + 5*n)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((n^2 + 5*n + (4*n^2 + 25*n + 25)*x + (5*n^2 + 35*n + 50)*x^2 + (2*n^2 + 15*n + 25)*x^3)*v[i - 1] + ((2*n + 5)*x + (6*n + 15)*x^2 + (6*n + 15)*x^3 + (2*n + 5)*x^4)*deriv(v[i - 1], x))/(n^2 + 5*n + (2*n^2 + 10*n)*x + (n^2 + 5*n)*x^2)); 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)

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-725543b0eb30a334
• 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-725543b0eb30a334
• 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-725543b0eb30a334

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

• preserver_image mobius, shift 0 e-725543b0eb30a334

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Coefficient triangle of certain polynomials N(5; m,x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-e968c429d4b85e0b root: theorem check

      Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

Related sequences ?

No candidate equalities or related sequences.

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