A334218 → D

OEIS matches ?

• A074143: row sums, 13 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:
   1: 0;
   2: 0;
   3: 2;
   4: 3,    6;
   5: 4,   32,     12;
   6: 5,  110,    165,     20;
   7: 6,  312,   1188,    624,     30;
   8: 7,  798,   6342,   8456,   1995,     42;
   9: 8, 1920,  28584,  77312,  47640,   5760,    56;
  10: 9, 4446, 115911, 562284, 702855, 231822, 15561, 72;

The row polynomials satisfy (n^2-5*n+6)*P(n,x) = -(n^4-9*n^3+28*n^2-36*n+16)*x^2*P(n-2,x) - ((n^3-5*n^2+8*n-4)*x^2-(n^3-5*n^2+8*n-4)*x^3)*d/dx P(n-2,x) + (n^2-4*n+3+(2*n^3-14*n^2+30*n-18)*x)*P(n-1,x) + ((n^2-4*n+3)*x-(n^2-4*n+3)*x^2)*d/dx P(n-1,x) for n >= 4, with P(1,x) = 0, P(2,x) = 0, P(3,x) = 2.

P[1, x_] := 0;
P[2, x_] := 0;
P[3, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^4 - 9*n^3 + 28*n^2 - 36*n + 16)*x^2*P[n - 2, x] - ((n^3 - 5*n^2 + 8*n - 4)*x^2 - (n^3 - 5*n^2 + 8*n - 4)*x^3)*D[P[n - 2, x], x] + (n^2 - 4*n + 3 + (2*n^3 - 14*n^2 + 30*n - 18)*x)*P[n - 1, x] + ((n^2 - 4*n + 3)*x - (n^2 - 4*n + 3)*x^2)*D[P[n - 1, x], x])/(n^2 - 5*n + 6)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 0; v[3] = 2; for(n = 4, N, my(i = n); v[i] = (-(n^4 - 9*n^3 + 28*n^2 - 36*n + 16)*x^2*v[i - 2] - ((n^3 - 5*n^2 + 8*n - 4)*x^2 - (n^3 - 5*n^2 + 8*n - 4)*x^3)*deriv(v[i - 2], x) + (n^2 - 4*n + 3 + (2*n^3 - 14*n^2 + 30*n - 18)*x)*v[i - 1] + ((n^2 - 4*n + 3)*x - (n^2 - 4*n + 3)*x^2)*deriv(v[i - 1], x))/(n^2 - 5*n + 6)); 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-5bbae1ad83a2b739
• 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-5bbae1ad83a2b739
• 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-5bbae1ad83a2b739
• 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-5bbae1ad83a2b739

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 derivative_polynomial, shift 0 e-5bbae1ad83a2b739

  Laguerre’s theorem on differential operators [SP14] polytool ?

  • premise (sequence): Triangle read by rows: T(n,k) is the number of permutations of 1..n arranged in a circle with exactly k descents.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang_derivative_lag e-9baf8deff42ffdae root: theorem check

      Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?

Related sequences ?

Same fingerprint (3)

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

• A173018 → θ+1
• A008292 → θ+1
• A008292 → θ+1 (shift +2)

References

1. [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
2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.