OFFSET
0,12
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1325
FORMULA
Column k is the k-th weigh transform of the all-ones sequence. The weigh transform of a sequence b has generating function Product_{i > 0} (1 + x^i)^b(i).
EXAMPLE
Array begins:
k=0 k=1 k=2 k=3 k=4 k=5 k=6
-----------------------------
n=0: 1 1 1 1 1 1 1
n=1: 1 1 1 1 1 1 1
n=2: 1 1 1 1 1 1 1
n=3: 1 2 3 4 5 6 7
n=4: 1 2 4 7 11 16 22
n=5: 1 3 7 14 25 41 63
n=6: 1 4 12 29 60 111 189
For example, the A(5,3) = 14 partitions are:
{{5}} {{1}}{{4}}
{{14}} {{2}}{{3}}
{{23}} {{1}}{{13}}
{{1}{4}} {{2}}{{12}}
{{2}{3}} {{1}}{{1}{3}}
{{1}{13}} {{2}}{{1}{2}}
{{2}{12}} {{1}}{{1}{12}}
MATHEMATICA
spl[n_, 0]:={n};
spl[n_, k_]:=Select[Join@@Table[Union[Sort/@Tuples[spl[#, k-1]&/@ptn]], {ptn, IntegerPartitions[n]}], UnsameQ@@#&];
Table[Length[spl[n-k, k]], {n, 0, 10}, {k, 0, n}]
PROG
(PARI)
WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}
M(n, k=n)={my(L=List(), v=vector(n, i, 1)); listput(L, concat([1], v)); for(j=1, k, v=WeighT(v); listput(L, concat([1], v))); Mat(Col(L))~}
{ my(A=M(7)); for(i=1, #A, print(A[i, ])) } \\ Andrew Howroyd, Dec 31 2019
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Dec 18 2019
STATUS
approved