A106800 → (θ+1)²

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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:
   1: 1;
   2: 1;
   3: 1,   4;
   4: 1,  12,    9;
   5: 1,  24,   63,    16;
   6: 1,  40,  225,   240,     25;
   7: 1,  60,  585,  1440,    775,     36;
   8: 1,  84, 1260,  5600,   7525,   2268,     49;
   9: 1, 112, 2394, 16800,  42525,  34776,   6223,    64;
  10: 1, 144, 4158, 42336, 173775, 279720, 148225, 16320, 81;

The row polynomials satisfy (n^2-4*n+4)*P(n,x) = -(n^2-3*n+2+(4*n^3-24*n^2+48*n-36)*x-(9*n^4-74*n^3+221*n^2-284*n+132)*x^2)*P(n-2,x) - (x-(4*n^3-20*n^2+32*n-11)*x^2-(4*n^4-46*n^3+167*n^2-244*n+124)*x^3)*d/dx P(n-2,x) + (x^3-(4*n^3-21*n^2+36*n-20)*x^4)*d^2/dx^2 P(n-2,x) + (2*n^2-7*n+6+(4*n^3-24*n^2+48*n-32)*x)*P(n-1,x) + (x-(4*n^3-20*n^2+32*n-16)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1.

P[1, x_] := 1;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 - 3*n + 2 + (4*n^3 - 24*n^2 + 48*n - 36)*x - (9*n^4 - 74*n^3 + 221*n^2 - 284*n + 132)*x^2)*P[n - 2, x] - (x - (4*n^3 - 20*n^2 + 32*n - 11)*x^2 - (4*n^4 - 46*n^3 + 167*n^2 - 244*n + 124)*x^3)*D[P[n - 2, x], x] + (x^3 - (4*n^3 - 21*n^2 + 36*n - 20)*x^4)*D[P[n - 2, x], {x, 2}] + (2*n^2 - 7*n + 6 + (4*n^3 - 24*n^2 + 48*n - 32)*x)*P[n - 1, x] + (x - (4*n^3 - 20*n^2 + 32*n - 16)*x^2)*D[P[n - 1, x], x])/(n^2 - 4*n + 4)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1; for(n = 3, N, my(i = n); v[i] = (-(n^2 - 3*n + 2 + (4*n^3 - 24*n^2 + 48*n - 36)*x - (9*n^4 - 74*n^3 + 221*n^2 - 284*n + 132)*x^2)*v[i - 2] - (x - (4*n^3 - 20*n^2 + 32*n - 11)*x^2 - (4*n^4 - 46*n^3 + 167*n^2 - 244*n + 124)*x^3)*deriv(v[i - 2], x) + (x^3 - (4*n^3 - 21*n^2 + 36*n - 20)*x^4)*deriv(deriv(v[i - 2], x), x) + (2*n^2 - 7*n + 6 + (4*n^3 - 24*n^2 + 48*n - 32)*x)*v[i - 1] + (x - (4*n^3 - 20*n^2 + 32*n - 16)*x^2)*deriv(v[i - 1], x))/(n^2 - 4*n + 4)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Proved properties

• Real-rooted. Every row P_n, n >= 1, 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-39092568b301666d
• Coefficients of one sign. Every nonzero row, n >= 1: 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-39092568b301666d
• Hadamard kernel (Malo). Rows P_n, n >= 1, 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-39092568b301666d
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-39092568b301666d

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-39092568b301666d

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

  • premise (sequence): Triangle of Stirling numbers of 2nd kind, S(n, n-k), n >= 0, 0 <= k <= n.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-2ab1dfe057e84c9d root: theorem check

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

Related sequences ?

Same fingerprint (1)

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

• A008278 → (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.