Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(x) = x + 3.

OEIS matches ?

• A001725: 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:
   1:      9,       24,         9;
   2:     27,      141,       141,         27;
   3:     81,      726,      1410,        726,          81;
   4:    243,     3471,     11406,      11406,        3471,         243;
   5:    729,    15828,     81327,     136872,       81327,       15828,         729;
   6:   2187,    69873,    533259,    1390521,     1390521,      533259,       69873,        2187;
   7:   6561,   301362,   3295152,   12609198,    19467294,    12609198,     3295152,      301362,       6561;
   8:  19683,  1277619,  19489380,  105311556,   237144642,   237144642,   105311556,    19489380,    1277619,     19683;
   9:  59049,  5346672, 111500709,  826763136,  2607816498,  3794314272,  2607816498,   826763136,  111500709,   5346672,    59049;
  10: 177147, 22154325, 621663609, 6187086615, 26522346846, 53824862658, 53824862658, 26522346846, 6187086615, 621663609, 22154325, 177147;

The row polynomials satisfy P(n,x) = (3+(n+3)*x)*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 9 + 24*x + 9*x^2.

P[1, x_] := 9 + 24*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[(3 + (n + 3)*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] = 9 + 24*x + 9*x^2; for(n = 2, N, my(i = n); v[i] = (3 + (n + 3)*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-27a325e06cca4d67
• 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-27a325e06cca4d67
• 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-27a325e06cca4d67
• 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-27a325e06cca4d67

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-27a325e06cca4d67 root: theorem check

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

Used in 3 proof steps

• e-418ae03de1c699ba gamma_polynomial (scale 1, shift 0) → gamma_polynomial of Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)*t(n-1,…
• e-52155003e339a993 obreschkoff_combination → obreschkoff_combination of Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)…
• e-52155003e339a993 obreschkoff_combination → obreschkoff_combination of Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)…

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