A144697: Triangle of 3-Eulerian numbers.

OEIS matches ?

• A001715: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 3 ?
• A144697: triangle, 45 terms agree, 9 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 3 ?

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:
  0: 1;
  1: 1,    3;
  2: 1,   10,      9;
  3: 1,   25,     67,      27;
  4: 1,   56,    326,     376,       81;
  5: 1,  119,   1314,    3134,     1909,      243;
  6: 1,  246,   4775,   20420,    25215,     9094,      729;
  7: 1,  501,  16293,  115105,   248595,   180639,    41479,    2187;
  8: 1, 1012,  53388,  590764,  2048710,  2575404,  1193548,  183412,   6561;
  9: 1, 2035, 170284, 2843548, 14969662, 29793394, 23807260, 7435036, 792697, 19683;

The row polynomials satisfy P(n,x) = (1+(n+2)*x)*P(n-1,x) + (x-x^2)*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[(1 + (n + 2)*x)*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], x]];
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] = (1 + (n + 2)*x)*v[i - 1] + (x - x^2)*deriv(v[i - 1], x)); 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)

Triangle begins:
   3: 1;
   4: 1,    3;
   5: 1,   10,      9;
   6: 1,   25,     67,      27;
   7: 1,   56,    326,     376,       81;
   8: 1,  119,   1314,    3134,     1909,      243;
   9: 1,  246,   4775,   20420,    25215,     9094,      729;
  10: 1,  501,  16293,  115105,   248595,   180639,    41479,    2187;
  11: 1, 1012,  53388,  590764,  2048710,  2575404,  1193548,  183412,   6561;
  12: 1, 2035, 170284, 2843548, 14969662, 29793394, 23807260, 7435036, 792697, 19683;

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

P[3, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + (n - 1)*x)*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], x]];
Table[PadRight[CoefficientList[P[n, x], x], n - 2], {n, 3, 12}] // Flatten

P(N) = {my(v = vector(N - 2)); v[1] = 1; for(n = 4, N, my(i = n - 2); v[i] = (1 + (n - 1)*x)*v[i - 1] + (x - x^2)*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - 2 - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i + 2, v[i])))};
T(12)

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-0b8ec5e4abd25b24
• 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-0b8ec5e4abd25b24
• 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-0b8ec5e4abd25b24
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 1
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-0b8ec5e4abd25b24

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 e-0b8ec5e4abd25b24 root: theorem check

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

Used in 2 proof steps

• e-b4867a0cc7e138f4 obreschkoff_combination → obreschkoff_combination of Triangle of 3-Eulerian numbers.
• e-b4867a0cc7e138f4 obreschkoff_combination → obreschkoff_combination of Triangle of 3-Eulerian numbers.

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