A062264 → x ↦ x/(x+1)

OEIS matches ?

• A049609: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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,   6;
  2: 1,  14,   28;
  3: 1,  24,  108,   120;
  4: 1,  36,  270,   660,    495;
  5: 1,  50,  550,  2200,   3575,    2002;
  6: 1,  66,  990,  5720,  15015,   18018,    8008;
  7: 1,  84, 1638, 12740,  47775,   91728,   86632,   31824;
  8: 1, 104, 2548, 25480, 127400,  346528,  519792,  403104,  125970;
  9: 1, 126, 3780, 47040, 299880, 1079568, 2279088, 2790720, 1831410, 497420;

The row polynomials satisfy (n^2+4*n+(2*n^2+8*n)*x+(n^2+4*n)*x^2)*P(n,x) = (n^2+4*n+(4*n^2+20*n+16)*x+(5*n^2+28*n+32)*x^2+(2*n^2+12*n+16)*x^3)*P(n-1,x) + ((2*n+4)*x+(6*n+12)*x^2+(6*n+12)*x^3+(2*n+4)*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 + 4*n + (4*n^2 + 20*n + 16)*x + (5*n^2 + 28*n + 32)*x^2 + (2*n^2 + 12*n + 16)*x^3)*P[n - 1, x] + ((2*n + 4)*x + (6*n + 12)*x^2 + (6*n + 12)*x^3 + (2*n + 4)*x^4)*D[P[n - 1, x], x])/(n^2 + 4*n + (2*n^2 + 8*n)*x + (n^2 + 4*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 + 4*n + (4*n^2 + 20*n + 16)*x + (5*n^2 + 28*n + 32)*x^2 + (2*n^2 + 12*n + 16)*x^3)*v[i - 1] + ((2*n + 4)*x + (6*n + 12)*x^2 + (6*n + 12)*x^3 + (2*n + 4)*x^4)*deriv(v[i - 1], x))/(n^2 + 4*n + (2*n^2 + 8*n)*x + (n^2 + 4*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-cfe6f0bc80bb6e33
• 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-cfe6f0bc80bb6e33
• 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-cfe6f0bc80bb6e33

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-cfe6f0bc80bb6e33

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Coefficient triangle of certain polynomials N(4; 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-c22f7d753efa18e6 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