A114656 → x ↦ x/(x+1)

OEIS matches ?

• A127846: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A059231: 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:   2,     3;
   3:   4,    14,     11;
   4:   8,    48,     84,      45;
   5:  16,   144,    416,     484,     197;
   6:  32,   400,   1680,    3160,    2750,     903;
   7:  64,  1056,   6000,   16120,   22380,   15546,    4279;
   8: 128,  2688,  19712,   70560,  138320,  151872,   87752,   20793;
   9: 256,  6656,  60928,  277760,  718144, 1104320, 1001728,  495368,  103049;
  10: 512, 16128, 179712, 1010688, 3298176, 6640704, 8385216, 6475824, 2798586, 518859;

The row polynomials satisfy (n+1+(2*n+2)*x+(n+1)*x^2)*P(n,x) = (2*n+2+(5*n+11)*x+(2*n+20)*x^2-(3*n-15)*x^3-(2*n-4)*x^4)*P(n-1,x) + (4*x+14*x^2+18*x^3+10*x^4+2*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[((2*n + 2 + (5*n + 11)*x + (2*n + 20)*x^2 - (3*n - 15)*x^3 - (2*n - 4)*x^4)*P[n - 1, x] + (4*x + 14*x^2 + 18*x^3 + 10*x^4 + 2*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] = ((2*n + 2 + (5*n + 11)*x + (2*n + 20)*x^2 - (3*n - 15)*x^3 - (2*n - 4)*x^4)*v[i - 1] + (4*x + 14*x^2 + 18*x^3 + 10*x^4 + 2*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-0286c892acd84d8a
• 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-0286c892acd84d8a
• 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-0286c892acd84d8a

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-0286c892acd84d8a

  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 >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-e6c0603bd7c28948 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/(x+1)
• A114687 → 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