A274498 → θ+1

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:   3;
  2:   3,   12;
  3:   9,   36;
  4:   9,   72,   108;
  5:  27,  216,   324;
  6:  27,  324,   972,   864;
  7:  81,  972,  2916,  2592;
  8:  81, 1296,  5832, 10368,  6480;
  9: 243, 3888, 17496, 31104, 19440;

The row polynomials satisfy (2+(2*n+3)*x+(n^2+n)*x^2)*P(n,x) = -(4*n-10+(6*n^2-18*n-3)*x+(2*n^3-11*n^2-9*n+12)*x^2-(2*n^3+4*n^2-2*n-4)*x^3)*P(n-1,x) + (2+(4*n+2)*x+(2*n^2+4*n-12)*x^2+(2*n^2-14*n-20)*x^3-(4*n^2+12*n+8)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n - 10 + (6*n^2 - 18*n - 3)*x + (2*n^3 - 11*n^2 - 9*n + 12)*x^2 - (2*n^3 + 4*n^2 - 2*n - 4)*x^3)*P[n - 1, x] + (2 + (4*n + 2)*x + (2*n^2 + 4*n - 12)*x^2 + (2*n^2 - 14*n - 20)*x^3 - (4*n^2 + 12*n + 8)*x^4)*D[P[n - 1, x], x])/(2 + (2*n + 3)*x + (n^2 + n)*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] = (-(4*n - 10 + (6*n^2 - 18*n - 3)*x + (2*n^3 - 11*n^2 - 9*n + 12)*x^2 - (2*n^3 + 4*n^2 - 2*n - 4)*x^3)*v[i - 1] + (2 + (4*n + 2)*x + (2*n^2 + 4*n - 12)*x^2 + (2*n^2 - 14*n - 20)*x^3 - (4*n^2 + 12*n + 8)*x^4)*deriv(v[i - 1], x))/(2 + (2*n + 3)*x + (n^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 (-inf, 0]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-493e6c3c99ee8db4
• 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-493e6c3c99ee8db4
• 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-493e6c3c99ee8db4

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-493e6c3c99ee8db4

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

  • premise (sequence): Triangle read by rows: T(n,k) is the number of ternary words of length n having degree of asymmetry equal to k (n>=0; 0…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • parent_provers periodic_product e-363ec3019ccc29a9 root: theorem check

      Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (1)

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

• A188440 → θ+1

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