A094638 → (θ+1)²

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: 1;
  1: 1,   4;
  2: 1,  12,   18;
  3: 1,  24,   99,     96;
  4: 1,  40,  315,    800,     600;
  5: 1,  60,  765,   3600,    6850,    4320;
  6: 1,  84, 1575,  11760,   40600,   63504,    35280;
  7: 1, 112, 2898,  31360,  169225,  472752,   640332,   322560;
  8: 1, 144, 4914,  72576,  561225, 2422224,  5788076,  7013376,  3265920;
  9: 1, 180, 7830, 151200, 1581825, 9695700, 35460320, 75052800, 83152656, 36288000;

The row polynomials satisfy (n-1)*P(n,x) = -(n+(4*n^2-4*n)*x-(9*n^3-18*n^2+9*n)*x^2)*P(n-2,x) + ((4*n^2-4*n)*x^2+(4*n^3-8*n^2+4*n)*x^3)*d/dx P(n-2,x) + (2*n-1+(4*n^2-4*n)*x)*P(n-1,x) - (4*n^2-4*n)*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.

P[0, x_] := 1;
P[1, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n + (4*n^2 - 4*n)*x - (9*n^3 - 18*n^2 + 9*n)*x^2)*P[n - 2, x] + ((4*n^2 - 4*n)*x^2 + (4*n^3 - 8*n^2 + 4*n)*x^3)*D[P[n - 2, x], x] + (2*n - 1 + (4*n^2 - 4*n)*x)*P[n - 1, x] - (4*n^2 - 4*n)*x^2*D[P[n - 1, x], x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n + (4*n^2 - 4*n)*x - (9*n^3 - 18*n^2 + 9*n)*x^2)*v[i - 2] + ((4*n^2 - 4*n)*x^2 + (4*n^3 - 8*n^2 + 4*n)*x^3)*deriv(v[i - 2], x) + (2*n - 1 + (4*n^2 - 4*n)*x)*v[i - 1] - (4*n^2 - 4*n)*x^2*deriv(v[i - 1], x))/(n - 1)); 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-85e78a9430ebb6ba
• 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-85e78a9430ebb6ba
• 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-85e78a9430ebb6ba
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-85e78a9430ebb6ba

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 euler_polynomial, shift 0 e-85e78a9430ebb6ba

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

  • premise (sequence): Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind A008276 (1…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-419a6747cb084155 root: theorem check

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

    • preserver_image reversal, shift 0 e-986a227a8a459030

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in incre…
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • theorem_check liu_wang e-5f21311bb67f8c1c root: theorem check

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

        • basis_map A007318 e-0b3ba9047fae59ed

          Geometric basis map (elementary) polytool ?

          • premise (sequence): Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • parent_provers root_domain e-edbf92ee2e446027 root: theorem check

              Parent-row provers of polytool polytool ?

        • parent_provers root_domain e-2e5c10abc86c264d root: theorem check

          Parent-row provers of polytool polytool ?

        • preserver_image reversal, shift 0 e-90e69a839bdbf74b

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind A008276 (1…
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • ↵ theorem_check liu_wang e-419a6747cb084155 (expanded above)
            • parent_provers paper_theorem e-b29f35705824bef6 root: theorem check

              Parent-row provers of polytool polytool ?

        • preserver_image affine, shift 0 e-1ba7a8eebd893648

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • ↵ parent_provers root_domain e-edbf92ee2e446027 (expanded above)
    • ↵ parent_provers paper_theorem e-b29f35705824bef6 (expanded above)

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. [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.