basis-map coefficients of A145596 → D+1

OEIS matches ?

• A006527: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A167875: 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:    1;
  1:    2,     2;
  2:    0,     8,    3;
  3:   -8,     8,   20,     4;
  4:   -5,   -40,   45,    40,      5;
  5:   30,   -90,  -90,   150,     70,     6;
  6:   28,   168, -525,   -56,    385,   112,    7;
  7: -112,   560,  168, -1848,    392,   840,  168,    8;
  8: -126,  -672, 3780, -2016,  -4620,  2016, 1638,  240,   9;
  9:  420, -2940,  840, 14280, -13440, -8400, 6300, 2940, 330, 10;

The row polynomials satisfy (n^4+2*n^3-3*n^2+20*n+(2*n^3+4*n^2-16*n)*x+(n^2+4*n)*x^2)*P(n,x) = -(2*n^4-4*n^3-6*n^2-16*n-16-(3*n^4-3*n^3+9*n^2+13*n-2)*x-(6*n^3-n^2-11*n-4)*x^2-(3*n^2+5*n+2)*x^3)*P(n-1,x) - (4*n^3+2*n^2+10*n+12-(6*n^3-5*n^2+15*n+26)*x+(2*n^3-11*n^2+9*n+22)*x^2+(4*n^2-6*n-10)*x^3+(2*n+2)*x^4)*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^4 - 4*n^3 - 6*n^2 - 16*n - 16 - (3*n^4 - 3*n^3 + 9*n^2 + 13*n - 2)*x - (6*n^3 - n^2 - 11*n - 4)*x^2 - (3*n^2 + 5*n + 2)*x^3)*P[n - 1, x] - (4*n^3 + 2*n^2 + 10*n + 12 - (6*n^3 - 5*n^2 + 15*n + 26)*x + (2*n^3 - 11*n^2 + 9*n + 22)*x^2 + (4*n^2 - 6*n - 10)*x^3 + (2*n + 2)*x^4)*D[P[n - 1, x], x])/(n^4 + 2*n^3 - 3*n^2 + 20*n + (2*n^3 + 4*n^2 - 16*n)*x + (n^2 + 4*n)*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] = (-(2*n^4 - 4*n^3 - 6*n^2 - 16*n - 16 - (3*n^4 - 3*n^3 + 9*n^2 + 13*n - 2)*x - (6*n^3 - n^2 - 11*n - 4)*x^2 - (3*n^2 + 5*n + 2)*x^3)*v[i - 1] - (4*n^3 + 2*n^2 + 10*n + 12 - (6*n^3 - 5*n^2 + 15*n + 26)*x + (2*n^3 - 11*n^2 + 9*n + 22)*x^2 + (4*n^2 - 6*n - 10)*x^3 + (2*n + 2)*x^4)*deriv(v[i - 1], x))/(n^4 + 2*n^3 - 3*n^2 + 20*n + (2*n^3 + 4*n^2 - 16*n)*x + (n^2 + 4*n)*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-2036104bf9bd843b

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-2036104bf9bd843b

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

  • premise (sequence): basis-map coefficients of Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 1)*binomial…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
    • preserver_image affine, shift 0 e-3babd9179153850c

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 1)*binomial…
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • theorem_check liu_wang e-c483eeee9c2638c9 root: theorem check

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

        • preserver_image reversal, shift 0 e-1d2c7b4a3cb3d72f

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): basis-map coefficients of Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 1)*binomial…
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • ↵ theorem_check liu_wang e-c483eeee9c2638c9 (expanded above)
            • basis_map A007318 e-fb63394f9047a486

              Geometric basis map (elementary) polytool ?

              • premise (sequence): basis-map coefficients of Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 1)*binomial…
                uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 1]
                no recorded property covers this requirement
        • ↵ basis_map A007318 e-fb63394f9047a486 (expanded above)
    • theorem_check three_term_recurrence e-8ef8162b0ed7a444 root: theorem check

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

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