|
%I #11 Mar 24 2017 13:49:27
%S 1,0,1,2,5,13,35,100,298,926,2995,10045,34871,125040,462283,1759340,
%T 6882479,27639252,113809750,479993898,2071411798,9138568984,
%U 41182104446,189418562699,888607018626,4248949407337,20695172225549,102617378820155,517728263280060
%N Number of collections of nonempty multisets of colored objects, where n is the number of objects plus the number of distinct colors.
%H Alois P. Heinz, <a href="/A258450/b258450.txt">Table of n, a(n) for n = 0..500</a>
%F a(n) = Sum_{i=0..floor(n/2)} A255903(n-i,i).
%e a(4) = 5: {{1},{1},{1}}, {{1},{1,1}}, {{1,1,1}}, {{1},{2}}, {{1,2}}.
%p with(numtheory):
%p A:= proc(n, k) option remember; `if`(n=0, 1, add(A(n-j, k)*
%p add(d*binomial(d+k-1, k-1), d=divisors(j)), j=1..n)/n)
%p end:
%p T:= (n, k)-> add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k):
%p a:= n-> add(T(n-i, i), i=0..n/2):
%p seq(a(n), n=0..30);
%t A[n_, k_] := A[n, k] = If[n==0, 1, Sum[A[n-j, k]*DivisorSum[j, #*Binomial[# +k-1, k-1]&], {j, 1, n}]/n];
%t T[n_, k_] := Sum[A[n, k-i]*(-1)^i*Binomial[k, i], {i, 0, k}];
%t a[n_] := Sum[T[n-i, i], {i, 0, n/2}];
%t Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Mar 24 2017, translated from Maple *)
%Y Antidiagonal sums of A255903.
%K nonn
%O 0,4
%A _Alois P. Heinz_, May 30 2015
|
