A062145 → x ↦ x/(x+1)

OEIS matches ?

• A049608: 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:
   1: 1;
   2: 1,   5;
   3: 1,  12,   21;
   4: 1,  21,   84,    84;
   5: 1,  32,  216,   480,    330;
   6: 1,  45,  450,  1650,   2475,   1287;
   7: 1,  60,  825,  4400,  10725,  12012,    5005;
   8: 1,  77, 1386, 10010,  35035,  63063,   56056,   19448;
   9: 1,  96, 2184, 20384,  95550, 244608,  346528,  254592,   75582;
  10: 1, 117, 3276, 38220, 229320, 779688, 1559376, 1813968, 1133730, 293930;

The row polynomials satisfy (n^2+n-2+(2*n^2+2*n-4)*x+(n^2+n-2)*x^2)*P(n,x) = (n^2+n-2+(4*n^2+7*n-2)*x+(5*n^2+11*n+2)*x^2+(2*n^2+5*n+2)*x^3)*P(n-1,x) + ((2*n+1)*x+(6*n+3)*x^2+(6*n+3)*x^3+(2*n+1)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + n - 2 + (4*n^2 + 7*n - 2)*x + (5*n^2 + 11*n + 2)*x^2 + (2*n^2 + 5*n + 2)*x^3)*P[n - 1, x] + ((2*n + 1)*x + (6*n + 3)*x^2 + (6*n + 3)*x^3 + (2*n + 1)*x^4)*D[P[n - 1, x], x])/(n^2 + n - 2 + (2*n^2 + 2*n - 4)*x + (n^2 + n - 2)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((n^2 + n - 2 + (4*n^2 + 7*n - 2)*x + (5*n^2 + 11*n + 2)*x^2 + (2*n^2 + 5*n + 2)*x^3)*v[i - 1] + ((2*n + 1)*x + (6*n + 3)*x^2 + (6*n + 3)*x^3 + (2*n + 1)*x^4)*deriv(v[i - 1], x))/(n^2 + n - 2 + (2*n^2 + 2*n - 4)*x + (n^2 + n - 2)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Proved properties

• Real-rooted. Every row P_n, n >= 1, 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-20496aa59e8ba6c2
• Coefficients of one sign. Every nonzero row, n >= 1: 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-20496aa59e8ba6c2
• Hadamard kernel (Malo). Rows P_n, n >= 1, 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-20496aa59e8ba6c2

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-20496aa59e8ba6c2

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Triangle read by rows: T(n, k) = [z^k] P(n, z) where P(n, z) = Sum_{k=0..n} binomial(n, k) * Pochhammer(n - k + c, k) *…
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-62efc507a1cd208a 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