OFFSET
0,8
LINKS
Alois P. Heinz, Rows n = 0..140, flattened
EXAMPLE
T(4,1) = 3: .o-o. .o-o. .o-o.
.| |. .| . .|\ .
.o-o. .o-o. .o o.
.
T(4,2) = 3: .o-o. .o-o. .o-o.
.|/ . .| . . .
.o o. .o o. .o-o.
.
T(5,1) = 4: .o-o-o. .o-o-o. .o-o-o. .o-o-o.
.| / . .| . .| | . . /| .
.o-o . .o-o . .o o . .o o .
.
T(5,2) = 5: .o-o o. .o-o o. .o-o o. .o o-o. .o o-o.
.| | . .| . .|\ . .|\ . .| .
.o-o . .o-o . .o o . .o-o . .o-o .
Triangle T(n,k) begins:
1;
0, 1;
0, 1, 1;
0, 2, 1, 1;
0, 3, 3, 1, 1;
0, 4, 5, 3, 1, 1;
0, 7, 10, 6, 3, 1, 1;
0, 12, 17, 12, 6, 3, 1, 1;
...
MATHEMATICA
b[n_] := b[n] = If[n <= 1, n, Sum[Sum[d*b[d], {d, Divisors[j]}]*b[n-j], {j, 1, n-1}]/(n-1)];
g[n_] := g[n] = If[n>2, 1, 0]+b[n]-(Sum [b[k]*b[n-k], {k, 0, n}] - If[Mod[n, 2] == 0, b[n/2], 0])/2;
p[n_, i_, t_] := p[n, i, t] = If[n<t, 0, If[n == t, 1, If[Min[i, t]<1, 0, Sum[Binomial[g[i]+j-1, j]*p[n-i*j, i-1, t-j], {j, 0, Min[n/i, t]}]]]];
T[n_, k_] := p[n, n, k] // FullSimplify;
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 04 2014, after Alois P. Heinz *)
PROG
(Python)
from sympy.core.cache import cacheit
from sympy import binomial, divisors
@cacheit
def b(n): return n if n<2 else sum([sum([d*b(d) for d in divisors(j)])*b(n - j) for j in range(1, n)])//(n - 1)
@cacheit
def g(n): return (1 if n>2 else 0) + b(n) - (sum(b(k)*b(n - k) for k in range(n + 1)) - (b(n//2) if n%2==0 else 0))//2
@cacheit
def p(n, i, t): return 0 if n<t else 1 if n==t else 0 if min(i, t)<1 else sum(binomial(g(i) + j - 1, j)*p(n - i*j, i - 1, t - j) for j in range(min(n//i, t) + 1))
def T(n, k): return p(n, n, k)
for n in range(21): print([T(n, k) for k in range(n + 1)]) # Indranil Ghosh, Aug 07 2017
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Aug 29 2012
STATUS
approved