A395454: Triangle read by rows: T(n, k) = Sum_{j=0..n} C(-j-1, -n-1)*E2(j, n-k), E2 the Eulerian numbers A201637.

OEIS matches ?

• A053871: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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: 0;
   3: 0,      2;
   4: 0,      6,       2;
   5: 0,     24,      34,       2;
   6: 0,    120,     324,      98,       2;
   7: 0,    720,    2988,    2096,     234,       2;
   8: 0,   5040,   28944,   34704,    9804,     514,      2;
   9: 0,  40320,  300816,  533592,  276004,   38856,   1082,    2;
  10: 0, 362880, 3371040, 8137512, 6535272, 1766008, 139940, 2226, 2;

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

P[1, x_] := 1;
P[2, x_] := 0;
P[n_, x_] := P[n, x] = Expand[(n - 2 + (n - 2)*x)*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], x] + (2*n - 4)*x*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 0; for(n = 3, N, my(i = n); v[i] = (n - 2 + (n - 2)*x)*v[i - 1] + (x - x^2)*deriv(v[i - 1], x) + (2*n - 4)*x*v[i - 2]); 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 >= 3, 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-0ca91705a707cafe
• Coefficients of one sign. Every nonzero row, n >= 3: 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-0ca91705a707cafe
• Hadamard kernel (Malo). Rows P_n, n >= 3, 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-0ca91705a707cafe
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 4
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-0ca91705a707cafe

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

• theorem_check liu_wang_derivative_lag e-0ca91705a707cafe root: theorem check

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

Related sequences ?

No candidate equalities or related sequences.

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