A132812 → x ↦ x/(x+1)

OEIS matches ?

• A050151: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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,   4;
  2:  3,  15,   15;
  3:  4,  36,   84,    56;
  4:  5,  70,  280,   420,    210;
  5:  6, 120,  720,  1800,   1980,    792;
  6:  7, 189, 1575,  5775,  10395,   9009,    3003;
  7:  8, 280, 3080, 15400,  40040,  56056,   40040,   11440;
  8:  9, 396, 5544, 36036, 126126, 252252,  288288,  175032,  43758;
  9: 10, 540, 9360, 76440, 343980, 917280, 1485120, 1432080, 755820, 167960;

The row polynomials satisfy (n^2+2*n+(2*n^2+4*n)*x+(n^2+2*n)*x^2)*P(n,x) = -(n^2+n+(2*n^2+2*n)*x+(n^2+n)*x^2)*P(n-2,x) + (2*n^2+3*n+1+(8*n^2+12*n+4)*x+(10*n^2+15*n+5)*x^2+(4*n^2+6*n+2)*x^3)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 4*x.

P[0, x_] := 1;
P[1, x_] := 2 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 + n + (2*n^2 + 2*n)*x + (n^2 + n)*x^2)*P[n - 2, x] + (2*n^2 + 3*n + 1 + (8*n^2 + 12*n + 4)*x + (10*n^2 + 15*n + 5)*x^2 + (4*n^2 + 6*n + 2)*x^3)*P[n - 1, x])/(n^2 + 2*n + (2*n^2 + 4*n)*x + (n^2 + 2*n)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^2 + n + (2*n^2 + 2*n)*x + (n^2 + n)*x^2)*v[i - 2] + (2*n^2 + 3*n + 1 + (8*n^2 + 12*n + 4)*x + (10*n^2 + 15*n + 5)*x^2 + (4*n^2 + 6*n + 2)*x^3)*v[i - 1])/(n^2 + 2*n + (2*n^2 + 4*n)*x + (n^2 + 2*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 [-1, 0]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-6a8f8db3c1596aaf
• 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-6a8f8db3c1596aaf
• 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-6a8f8db3c1596aaf

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-6a8f8db3c1596aaf

  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]
        • 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 (6)

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

• narayana → basis A007318
• A063007 → D
• A132812 → x ↦ x+1
• narayana → x ↦ x/(x+1)
• A033282: Triangle read by rows: T(n, k) is the number of diagonal dissections of a convex n-gon into k+1 regions.
• A126216: Triangle read by rows: T(n,k) is the number of Schroeder paths of semilength n containing exactly k peaks but …

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