A114687 → x ↦ x/(x+1)

OEIS matches ?

• A003645: 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:
  0: 1;
  1: 1,  3;
  2: 1,  8,   11;
  3: 1, 15,   51,    45;
  4: 1, 24,  146,   304,    197;
  5: 1, 35,  330,  1190,   1765,    903;
  6: 1, 48,  645,  3520,   8835,  10128,    4279;
  7: 1, 63, 1141,  8715,  32515,  62013,   57799,   20793;
  8: 1, 80, 1876, 19040,  98014, 274064,  419524,  329024,  103049;
  9: 1, 99, 2916, 37884, 256158, 977130, 2169300, 2766060, 1871145, 518859;

The row polynomials satisfy (n+2+(2*n+4)*x+(n+2)*x^2)*P(n,x) = (n+2+(7*n+8)*x+(13*n+8)*x^2+9*n*x^3+(2*n-2)*x^4)*P(n-1,x) + (2*x+4*x^2-4*x^4-2*x^5)*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 + (7*n + 8)*x + (13*n + 8)*x^2 + 9*n*x^3 + (2*n - 2)*x^4)*P[n - 1, x] + (2*x + 4*x^2 - 4*x^4 - 2*x^5)*D[P[n - 1, x], x])/(n + 2 + (2*n + 4)*x + (n + 2)*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 + (7*n + 8)*x + (13*n + 8)*x^2 + 9*n*x^3 + (2*n - 2)*x^4)*v[i - 1] + (2*x + 4*x^2 - 4*x^4 - 2*x^5)*deriv(v[i - 1], x))/(n + 2 + (2*n + 4)*x + (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 - 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-c838eb74321cec2f
• 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-c838eb74321cec2f
• 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-c838eb74321cec2f

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-c838eb74321cec2f

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also …
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-0f37534727a0e0cd root: theorem check

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

Related sequences ?

Same fingerprint (2)

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

• A114655 → x ↦ x+1
• A114656 → 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