A132812 → γ

OEIS matches ?

• A005717: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 1 ?

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:  3,   3;
  3:  4,  12;
  4:  5,  30,   10;
  5:  6,  60,   60;
  6:  7, 105,  210,   35;
  7:  8, 168,  560,  280;
  8:  9, 252, 1260, 1260,  126;
  9: 10, 360, 2520, 4200, 1260;

The row polynomials satisfy (n+2)*P(n,x) = (3*n+3)*P(n-1,x) - (3*n-(4*n+4)*x)*P(n-2,x) + (n-1-(4*n-4)*x)*P(n-3,x) for n >= 3, with P(0,x) = 1, P(1,x) = 2, P(2,x) = 3 + 3*x.

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

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2; v[3] = 3 + 3*x; for(n = 3, N, my(i = n + 1); v[i] = ((3*n + 3)*v[i - 1] - (3*n - (4*n + 4)*x)*v[i - 2] + (n - 1 - (4*n - 4)*x)*v[i - 3])/(n + 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, 0]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-cc6de87a980d771e
• Coefficients of one sign. Every nonzero row, n >= 0: 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-cc6de87a980d771e
• Hadamard kernel (Malo). Rows P_n, n >= 0, 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-cc6de87a980d771e

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

• gamma_polynomial scale 1, shift 0 e-cc6de87a980d771e

  Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?

  • 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]
        • 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 (expanded above)
            • 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)
                • 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)
    • ↵ preserver_image reversal, shift 0 e-fcd77baa2d5e1258 (expanded above)
    • ↵ theorem_check liu_wang e-f8a41d41829f0ed1 (expanded above)

Related sequences ?

Same fingerprint (4)

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

• A011973 → ⊙ C(n,k) → D
• A055151: Triangular array of Motzkin polynomial coefficients.
• A059344 → ⊙ 1/k! (shift -1) → D
• narayana → γ

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