Triangle T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with p(1, x) = x/(1-x)^2, p(2, x) = x*(1+x)/(1-x)^3, and p(3, x) = x*(1+8*x+x^2)/(1-x)^4, read by rows.

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:
   1: 1,    8,      1;
   2: 1,   19,     19,        1;
   3: 1,   42,    114,       42,         1;
   4: 1,   89,    510,      510,        89,         1;
   5: 1,  184,   1975,     4080,      1975,       184,         1;
   6: 1,  375,   7029,    26195,     26195,      7029,       375,         1;
   7: 1,  758,  23712,   146954,    261950,    146954,     23712,       758,        1;
   8: 1, 1525,  77200,   753800,   2191474,   2191474,    753800,     77200,     1525,      1;
   9: 1, 3060, 245325,  3632800,  16233970,  26297688,  16233970,   3632800,   245325,   3060,    1;
  10: 1, 6131, 766575, 16739125, 110232250, 271423918, 271423918, 110232250, 16739125, 766575, 6131, 1;

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 >= 2, with P(1,x) = 1 + 8*x + x^2.

P[1, x_] := 1 + 8*x + x^2;
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[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1 + 8*x + x^2; for(n = 2, N, my(i = n); 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};
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-7ffc7a4c532b7fe3
• 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-7ffc7a4c532b7fe3, e-9eddb58049332ffc
• 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-7ffc7a4c532b7fe3, e-9eddb58049332ffc
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 2
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-7ffc7a4c532b7fe3
• 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-9eddb58049332ffc

Proof chains

2 incoming proof steps. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

• parent_provers root_domain e-9eddb58049332ffc root: theorem check

  Parent-row provers of polytool polytool ?

• theorem_check liu_wang e-7ffc7a4c532b7fe3 root: theorem check

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

Used in 3 proof steps

• e-b61cfbcaf1cd183b gamma_polynomial (scale 1, shift 0) → gamma_polynomial of Triangle T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^…
• e-8df0a750c7bea2d2 obreschkoff_combination → obreschkoff_combination of Triangle T(n, k) = coefficients of ( t(n, x) ) where t(n, x) =…
• e-8df0a750c7bea2d2 obreschkoff_combination → obreschkoff_combination of Triangle T(n, k) = coefficients of ( t(n, x) ) where t(n, x) =…

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