basis-map coefficients of A132812 → D

OEIS matches ?

• A006002: 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:    0;
  1:    2;
  2:    3,      6;
  3:  -12,     24,    12;
  4:  -20,    -40,    90,    20;
  5:   60,   -240,     0,   240,     30;
  6:  105,    210, -1155,   420,    525,    42;
  7: -280,   1680,  -840, -3360,   1960,  1008,    56;
  8: -504,  -1008,  9828, -9576,  -6300,  6048,  1764,   72;
  9: 1260, -10080, 10080, 30240, -44100, -5040, 15120, 2880, 90;

The row polynomials satisfy (n^3+n^2-2*n)*P(n,x) = -(2*n^3-2*n^2-8*n-4-(3*n^3+2*n^2-3*n-2)*x)*P(n-1,x) - (4*n^2+6*n+2-(6*n^2+9*n+3)*x+(2*n^2+3*n+1)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.

P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^3 - 2*n^2 - 8*n - 4 - (3*n^3 + 2*n^2 - 3*n - 2)*x)*P[n - 1, x] - (4*n^2 + 6*n + 2 - (6*n^2 + 9*n + 3)*x + (2*n^2 + 3*n + 1)*x^2)*D[P[n - 1, x], x])/(n^3 + n^2 - 2*n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n^3 - 2*n^2 - 8*n - 4 - (3*n^3 + 2*n^2 - 3*n - 2)*x)*v[i - 1] - (4*n^2 + 6*n + 2 - (6*n^2 + 9*n + 3)*x + (2*n^2 + 3*n + 1)*x^2)*deriv(v[i - 1], x))/(n^3 + n^2 - 2*n)); 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-d29f61265fea8d1e

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-d29f61265fea8d1e

  Laguerre’s theorem on differential operators [SP14] polytool ?

  • premise (sequence): basis-map coefficients of Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
    • preserver_image affine, shift 0 e-676bb32d6a1b9c64

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1).
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • preserver_image euler_polynomial, shift 1 e-041eea8fc0a7af05

          Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?

          • premise (sequence): narayana
            uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
            • theorem_check three_term_recurrence e-46a461774e271ec3 root: theorem check

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

            • preserver_image reversal, shift 0 e-29ef477df4172e22

              Affine, reversal and Möbius substitutions (elementary) polytool ?

              • premise (sequence): narayana
                uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
                • ↵ theorem_check three_term_recurrence e-46a461774e271ec3 (expanded above)
        • basis_map A007318 e-a32917d44ac940a7

          Geometric basis map (elementary) polytool ?

          • premise (sequence): basis-map coefficients of Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1).
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 1]
            no recorded property covers this requirement
        • preserver_image reversal, shift 0 e-fcd77baa2d5e1258

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1).
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • ↵ preserver_image euler_polynomial, shift 1 e-041eea8fc0a7af05 (expanded above)
            • ↵ basis_map A007318 e-a32917d44ac940a7 (expanded above)
            • theorem_check liu_wang e-f8a41d41829f0ed1 root: theorem check

              Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

        • ↵ theorem_check liu_wang e-f8a41d41829f0ed1 (expanded above)
    • theorem_check three_term_recurrence e-72ab2c9b35ff70ef root: theorem check

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

Related sequences ?

No candidate equalities or related sequences.

References

1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
3. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023