A156366: Triangle of coefficients of p(n, x) where p(n, x) is defined as p(n, x) = (1-3*x)^(n+1)*PolyLog(-n, 3*x)/(3*x), read by rows.

OEIS matches ?

• A122704: row sums, 15 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:
  0: 1,    12,       9;
  1: 1,    33,      99,        27;
  2: 1,    78,     594,       702,         81;
  3: 1,   171,    2718,      8154,       4617,         243;
  4: 1,   360,   10719,     65232,      96471,       29160,          729;
  5: 1,   741,   38637,    421713,    1265139,     1043199,       180063,         2187;
  6: 1,  1506,  131472,   2382318,   12651390,    21440862,     10649232,      1097874,        6561;
  7: 1,  3039,  430560,  12290184,  106138674,   318416022,    331834968,    104626080,     6646293,      19683;
  8: 1,  6108, 1373733,  59494176,  788787234,  3820992264,   7099085106,   4819028256,  1001451357,   40074588,     59049;
  9: 1, 12249, 4304439, 275067495, 5371796394, 39490485498, 118471456494, 145038502638, 66841401285, 9413808093, 241097067, 177147;

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

P[0, x_] := 1 + 12*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[(1 + (3*n + 6)*x)*P[n - 1, x] + (x - 3*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 + 12*x + 9*x^2; for(n = 1, N, my(i = n + 1); v[i] = (1 + (3*n + 6)*x)*v[i - 1] + (x - 3*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)

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-e495670ea063ae0d
• 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-e495670ea063ae0d
• 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-e495670ea063ae0d
• 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-e495670ea063ae0d

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-e495670ea063ae0d root: theorem check

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

Used in 2 proof steps

• e-4796775b805410d0 obreschkoff_combination → obreschkoff_combination of Triangle of coefficients of p(n, x) where p(n, x) is defined a…
• e-4796775b805410d0 obreschkoff_combination → obreschkoff_combination of Triangle of coefficients of p(n, x) where p(n, x) is defined a…

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