OFFSET
1,4
COMMENTS
Number of compositions of n-1 into distinct parts if there are a(i) kinds of part i. a(6) = 13: 5, 5', 5'', 5''', 5'''', 41, 4'1, 4''1, 14, 14', 14'', 32, 23.
LINKS
FORMULA
Shifts left under "AFK" (ordered, size, unlabeled) transform
MAPLE
b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0,
b(n, i-1, p) +`if`(i>n, 0, a(i)*b(n-i, i-1, p+1))))
end:
a:= n-> b(n-1$2, 0):
seq(a(n), n=1..40); # Alois P. Heinz, Sep 05 2015
MATHEMATICA
b[n_, i_, p_] := b[n, i, p] = If[n==0, p!, If[i<1, 0, b[n, i-1, p] + If[ i>n, 0, a[i]*b[n-i, i-1, p+1]]]]; a[n_] := b[n-1, n-1, 0]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Feb 24 2016, after Alois P. Heinz *)
PROG
(PARI)
AFK(v)={apply(p->subst(serlaplace(y^0*p), y, 1), Vec(prod(k=1, #v, 1 + v[k]*x^k*y + O(x*x^#v))))}
seq(n)={my(v=[1]); for(i=1, n, v=AFK(v)); v} \\ Andrew Howroyd, Jun 21 2018
CROSSREFS
KEYWORD
nonn,eigen
AUTHOR
EXTENSIONS
More terms from Alois P. Heinz, Sep 05 2015
STATUS
approved