A060920 → D

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:     0;
  1:     1;
  2:     5,     2;
  3:    20,    18,      3;
  4:    71,   102,     42,      4;
  5:   235,   466,    315,     80,     5;
  6:   744,  1884,   1782,    760,   135,     6;
  7:  2285,  7044,   8580,   5180,  1575,   210,    7;
  8:  6865, 24946,  37206,  29140, 12670,  2940,  308,   8;
  9: 20284, 84894, 149889, 144488, 82035, 27468, 5082, 432, 9;

The row polynomials satisfy (4*n^2-10*n+6)*P(n,x) = (16*n^2-32*n+10+(12*n^2-26*n+8)*x)*P(n-1,x) - (10*n-5+(18*n-9)*x+(8*n-4)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.

P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((16*n^2 - 32*n + 10 + (12*n^2 - 26*n + 8)*x)*P[n - 1, x] - (10*n - 5 + (18*n - 9)*x + (8*n - 4)*x^2)*D[P[n - 1, x], x])/(4*n^2 - 10*n + 6)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((16*n^2 - 32*n + 10 + (12*n^2 - 26*n + 8)*x)*v[i - 1] - (10*n - 5 + (18*n - 9)*x + (8*n - 4)*x^2)*deriv(v[i - 1], x))/(4*n^2 - 10*n + 6)); 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-2dfe8925c3432b87

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 derivative_polynomial, shift 0 e-2dfe8925c3432b87

  Laguerre’s theorem on differential operators [SP14] 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
3. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023