OFFSET
1,2
LINKS
Alois P. Heinz, Rows n = 1..45, flattened
FORMULA
EXAMPLE
Triangle T(n,k) begins:
1;
2, 1;
6, 3, 4;
32, 12, 16, 38;
320, 80, 80, 190, 728;
6144, 960, 640, 1140, 4368, 26704;
229376, 21504, 8960, 10640, 30576, 186928, 1866256;
...
MAPLE
b:= proc(n) option remember; `if`(n=0, 1, 2^(n*(n-1)/2)-
add(k*binomial(n, k)* 2^((n-k)*(n-k-1)/2)*b(k), k=1..n-1)/n)
end:
T:= (n, k)-> binomial(n, k)*b(k)*2^((n-k)*(n-k-1)/2):
seq(seq(T(n, k), k=1..n), n=1..10); # Alois P. Heinz, Aug 26 2013
MATHEMATICA
nn = 9; f[list_] := Select[list, # > 0 &]; g = Sum[2^Binomial[n, 2] x^n/n!, {n, 0, nn}]; a = Drop[Range[0, nn]! CoefficientList[Series[Log[g] + 1, {x, 0, nn}], x], 1]; Map[f, Drop[Transpose[Table[Range[0, nn]! CoefficientList[Series[a[[n]] x^n/n! g, {x, 0, nn}], x], {n, 1, nn}]], 1]] // Grid
PROG
(Magma)
function b(n) // b = A001187
if n eq 0 then return 1;
else return 2^Binomial(n, 2) - (&+[Binomial(n-1, j-1)*2^Binomial(n-j, 2)*b(j): j in [0..n-1]]);
end if; return b;
end function;
A223894:= func< n, k | Binomial(n, k)*2^Binomial(n-k, 2)*b(k) >;
[A223894(n, k): k in [1..n], n in [1..12]]; // G. C. Greubel, Oct 03 2022
(SageMath)
@CachedFunction
def b(n): # b = A001187
if (n==0): return 1
else: return 2^binomial(n, 2) - sum(binomial(n-1, j-1)*2^binomial(n-j, 2)*b(j) for j in range(n))
def A223894(n, k): return binomial(n, k)*2^binomial(n-k, 2)*b(k)
flatten([[A223894(n, k) for k in range(1, n+1)] for n in range(1, 13)]) # G. C. Greubel, Oct 03 2022
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Geoffrey Critzer, Mar 28 2013
STATUS
approved