A085478 → x ↦ x/(x+1)

OEIS matches ?

• A060920: reversed triangle, 55 terms agree, 10 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 0 ?
• A007583: 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,  2;
   3: 1,  5,   5;
   4: 1,  9,  20,   13;
   5: 1, 14,  51,   71,    34;
   6: 1, 20, 105,  233,   235,    89;
   7: 1, 27, 190,  594,   942,   744,   233;
   8: 1, 35, 315, 1295,  2860,  3522,  2285,   610;
   9: 1, 44, 490, 2534,  7285, 12402, 12473,  6865,  1597;
  10: 1, 54, 726, 4578, 16407, 36122, 49963, 42447, 20284, 4181;

The row polynomials satisfy (2*n-3+(4*n-6)*x+(2*n-3)*x^2)*P(n,x) = (2*n-3+(3*n-2)*x-(5*n-15)*x^2-(11*n-24)*x^3-(5*n-10)*x^4)*P(n-1,x) + (4*x+17*x^2+27*x^3+19*x^4+5*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 - 3 + (3*n - 2)*x - (5*n - 15)*x^2 - (11*n - 24)*x^3 - (5*n - 10)*x^4)*P[n - 1, x] + (4*x + 17*x^2 + 27*x^3 + 19*x^4 + 5*x^5)*D[P[n - 1, x], x])/(2*n - 3 + (4*n - 6)*x + (2*n - 3)*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 - 3 + (3*n - 2)*x - (5*n - 15)*x^2 - (11*n - 24)*x^3 - (5*n - 10)*x^4)*v[i - 1] + (4*x + 17*x^2 + 27*x^3 + 19*x^4 + 5*x^5)*deriv(v[i - 1], x))/(2*n - 3 + (4*n - 6)*x + (2*n - 3)*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)

Triangle begins:
  0:    1;
  1:    2,     1;
  2:    5,     5,     1;
  3:   13,    20,     9,     1;
  4:   34,    71,    51,    14,     1;
  5:   89,   235,   233,   105,    20,     1;
  6:  233,   744,   942,   594,   190,    27,    1;
  7:  610,  2285,  3522,  2860,  1295,   315,   35,   1;
  8: 1597,  6865, 12473, 12402,  7285,  2534,  490,  44,  1;
  9: 4181, 20284, 42447, 49963, 36122, 16407, 4578, 726, 54, 1;

Row n lists the coefficients of P(n,x) in descending powers of x, where (2*n-1+(4*n-2)*x+(2*n-1)*x^2)*P(n,x) = (2*n-1+(3*n+1)*x-(5*n-10)*x^2-(11*n-13)*x^3-(5*n-5)*x^4)*P(n-1,x) + (4*x+17*x^2+27*x^3+19*x^4+5*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[((2*n - 1 + (3*n + 1)*x - (5*n - 10)*x^2 - (11*n - 13)*x^3 - (5*n - 5)*x^4)*P[n - 1, x] + (4*x + 17*x^2 + 27*x^3 + 19*x^4 + 5*x^5)*D[P[n - 1, x], x])/(2*n - 1 + (4*n - 2)*x + (2*n - 1)*x^2)]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((2*n - 1 + (3*n + 1)*x - (5*n - 10)*x^2 - (11*n - 13)*x^3 - (5*n - 5)*x^4)*v[i - 1] + (4*x + 17*x^2 + 27*x^3 + 19*x^4 + 5*x^5)*deriv(v[i - 1], x))/(2*n - 1 + (4*n - 2)*x + (2*n - 1)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); 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 >= 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-9b55c08b8bb1817c
• 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-9b55c08b8bb1817c
• 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-9b55c08b8bb1817c

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-9b55c08b8bb1817c

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Triangle read by rows: T(n, k) = binomial(n + k, 2*k).
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-4, 0)
    • theorem_check three_term_recurrence e-59d957f5dca11392 root: theorem check

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

Related sequences ?

Same fingerprint (3)

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

• A060920 → reverse
• A060920: Bisection of Fibonacci triangle A037027: even-indexed members of column sequences of A037027 (not counting lea…
• A122076 → D

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