A060920 → x ↦ x/(x+1)

OEIS matches ?

No OEIS matches.

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:    2,     3;
  2:    5,    15,     11;
  3:   13,    59,     88,      43;
  4:   34,   207,    468,     465,     171;
  5:   89,   680,   2063,    3104,    2314,     683;
  6:  233,  2142,   8157,   16462,   18559,   11075,    2731;
  7:  610,  6555,  30042,   76095,  115005,  103665,   51580,   10923;
  8: 1597, 19641, 105244,  320837,  608455,  734861,  551798,  235445,   43691;
  9: 4181, 57913, 355235, 1266248, 2889997, 4378793, 4403439, 2833422, 1058286, 174763;

The row polynomials satisfy (2*n-1+(4*n-2)*x+(2*n-1)*x^2)*P(n,x) = (13*n-11+(59*n-52)*x+(97*n-89)*x^2+(69*n-66)*x^3+(18*n-18)*x^4)*P(n-1,x) - (5+34*x+90*x^2+116*x^3+73*x^4+18*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[((13*n - 11 + (59*n - 52)*x + (97*n - 89)*x^2 + (69*n - 66)*x^3 + (18*n - 18)*x^4)*P[n - 1, x] - (5 + 34*x + 90*x^2 + 116*x^3 + 73*x^4 + 18*x^5)*D[P[n - 1, x], x])/(2*n - 1 + (4*n - 2)*x + (2*n - 1)*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] = ((13*n - 11 + (59*n - 52)*x + (97*n - 89)*x^2 + (69*n - 66)*x^3 + (18*n - 18)*x^4)*v[i - 1] - (5 + 34*x + 90*x^2 + 116*x^3 + 73*x^4 + 18*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};
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 (-inf, +inf)
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-2d8eb3c5691a59ac

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-2d8eb3c5691a59ac

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Bisection of Fibonacci triangle A037027: even-indexed members of column sequences of A037027 (not counting leading zero…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
    • theorem_check three_term_recurrence e-1b409596bf82290e root: theorem check

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

    • preserver_image affine, shift 0 e-f409e7a544612dfe

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangular array binomial(2*n-k, k), k=0..n, n >= 0.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • theorem_check three_term_recurrence e-abbe6e2e6a5c7279 root: theorem check

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

        • gamma_polynomial scale 1, shift 0 e-313e6d9276f9a630

          Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?

          • premise (sequence): Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle.
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • basis_map A008288 e-80180a24956ab79f

              Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?

              • premise (sequence): basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…
                uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in [0, 0]
                no recorded property covers this requirement
            • basis_map A008288 e-a8f2d1946d8f3a6f

              Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?

              • premise (sequence): basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…
                uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in [0, 0]
                no recorded property covers this requirement
            • basis_map A008288 e-054f97e0823e3ab6

              Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?

              • premise (sequence): basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…
                uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in [0, 0]
                no recorded property covers this requirement
            • basis_map A008288 e-912230b718f958a6

              Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?

              • premise (sequence): basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…
                uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in [0, 0]
                no recorded property covers this requirement
    • basis_map A007318 e-5dc8ce8b8b471ab9

      Geometric basis map (elementary) polytool ?

      • premise (sequence): Triangular array binomial(2*n-k, k), k=0..n, n >= 0.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ theorem_check three_term_recurrence e-abbe6e2e6a5c7279 (expanded above)
        • ↵ gamma_polynomial scale 1, shift 0 e-313e6d9276f9a630 (expanded above)

Related sequences ?

No candidate equalities or related sequences.

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