A063007 → x ↦ x/(x+1)

OEIS matches ?

• A069835: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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,  3;
   3: 1,  8,   13;
   4: 1, 15,   57,    63;
   5: 1, 24,  156,   384,    321;
   6: 1, 35,  340,  1380,   2505,    1683;
   7: 1, 48,  645,  3800,  11145,   16008,    8989;
   8: 1, 63, 1113,  8855,  37065,   84903,  100849,   48639;
   9: 1, 80, 1792, 18368, 102340,  330848,  621040,  628736,  265729;
  10: 1, 99, 2736, 34944, 247716, 1055628, 2774352, 4408848, 3888657, 1462563;

The row polynomials satisfy (n-1+(2*n-2)*x+(n-1)*x^2)*P(n,x) = (n-1+(3*n-1)*x-(n-9)*x^2-(7*n-17)*x^3-(4*n-8)*x^4)*P(n-1,x) + (2*x+10*x^2+18*x^3+14*x^4+4*x^5)*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 - 1 + (3*n - 1)*x - (n - 9)*x^2 - (7*n - 17)*x^3 - (4*n - 8)*x^4)*P[n - 1, x] + (2*x + 10*x^2 + 18*x^3 + 14*x^4 + 4*x^5)*D[P[n - 1, x], x])/(n - 1 + (2*n - 2)*x + (n - 1)*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 - 1 + (3*n - 1)*x - (n - 9)*x^2 - (7*n - 17)*x^3 - (4*n - 8)*x^4)*v[i - 1] + (2*x + 10*x^2 + 18*x^3 + 14*x^4 + 4*x^5)*deriv(v[i - 1], x))/(n - 1 + (2*n - 2)*x + (n - 1)*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-7f33caeda894b198
• 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-7f33caeda894b198
• 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-7f33caeda894b198

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-7f33caeda894b198

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): T(n,k) = binomial(n,k)*binomial(n+k,k), 0 <= k <= n, triangle read by rows.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-1, 0)
    • theorem_check three_term_recurrence e-93f30859d3148f4f root: theorem check

      Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

Related sequences ?

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

• A104684 → basis A007318

References

1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
2. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023