A178821 → (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:   5,    20;
  2:  15,   120,    135;
  3:  35,   420,    945,    560;
  4:  70,  1120,   3780,   4480,    1750;
  5: 126,  2520,  11340,  20160,   15750,    4536;
  6: 210,  5040,  28350,  67200,   78750,   45360,   10290;
  7: 330,  9240,  62370, 184800,  288750,  249480,  113190,   21120;
  8: 495, 15840, 124740, 443520,  866250,  997920,  679140,  253440,  40095;
  9: 715, 25740, 231660, 960960, 2252250, 3243240, 2942940, 1647360, 521235, 71500;

The row polynomials satisfy (n+(3*n^2-n)*x+n^3*x^2)*P(n,x) = (n+4+(3*n^2+15*n+12)*x+(n^3+9*n^2+23*n+12)*x^2+(n^3+6*n^2+9*n+4)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n + 4 + (3*n^2 + 15*n + 12)*x + (n^3 + 9*n^2 + 23*n + 12)*x^2 + (n^3 + 6*n^2 + 9*n + 4)*x^3)*P[n - 1, x])/(n + (3*n^2 - n)*x + n^3*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] = ((n + 4 + (3*n^2 + 15*n + 12)*x + (n^3 + 9*n^2 + 23*n + 12)*x^2 + (n^3 + 6*n^2 + 9*n + 4)*x^3)*v[i - 1])/(n + (3*n^2 - n)*x + n^3*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, 0]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-7905a75bef9f2dad
• 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-7905a75bef9f2dad
• 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-7905a75bef9f2dad
• 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-7905a75bef9f2dad

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-7905a75bef9f2dad

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

  • premise (sequence): Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-20c79d8930f25d5b root: theorem check

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

    • preserver_image reversal, shift 0 e-12f9ed6b8198f9ce

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ theorem_check liu_wang e-20c79d8930f25d5b (expanded above)

Related sequences ?

Same fingerprint (12)

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

• basis-map coefficients of A113214, shifted → (θ+1)²
• A028326 → (θ+1)²
• A038208 → (θ+1)²
• A007318 → ×(1+2n+n²) → (θ+1)²
• A178820 → (θ+1)²
• A007318 → (θ+1)² → reverse
• A007318 → D → (θ+1)²
• A178822 → (θ+1)²
• A135838 → (θ+1)²
• A125092: Triangle read by rows: T(n,k) = (k+1)^2*binomial(n,k) (0 <= k <= n).
• A344565 → (θ+1)²
• A085880 → (θ+1)²

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.