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A373306
Sum over all complete compositions of n of the element multiset size.
3
0, 1, 2, 7, 13, 30, 73, 157, 345, 743, 1650, 3517, 7593, 16120, 34294, 72683, 153475, 323293, 679231, 1423721, 2977692, 6218395, 12959249, 26970243, 56037071, 116280086, 240953162, 498719275, 1031029386, 2129266321, 4392871427, 9054428894, 18645998093
OFFSET
0,3
COMMENTS
A complete composition of n has element set [k] with k<=n (without gaps).
LINKS
FORMULA
G.f.: Sum_{k>0} d/dy C({1..k},x,y)|y = 1 where C({s},x,y) = Sum_{i in {s}} (C({s}-{i},x,y)*y*x^i)/(1 - Sum_{i in {s}} (y*x^i)) with C({},x,y) = 1. - John Tyler Rascoe, Jun 18 2024
EXAMPLE
a(1) = 1: 1.
a(2) = 2: 11.
a(3) = 7 = 2 + 2 + 3: 12, 21, 111.
a(4) = 13 = 3 + 3 + 3 + 4: 112, 121, 211, 1111.
a(5) = 30 = 3*3 + 4*4 + 5: 122, 212, 221, 1112, 1121, 1211, 2111, 11111.
MAPLE
b:= proc(n, i, t) option remember; `if`(n=0, `if`(i=0, [t!, 0], 0),
`if`(i<1 or n<i*(i+1)/2, 0, add((p-> p+[0, p[1]]*j)(
b(n-i*j, i-1, t+j)/j!), j=1..n/i)))
end:
a:= n-> add(b(n, k, 0)[2], k=0..floor((sqrt(1+8*n)-1)/2)):
seq(a(n), n=0..32);
MATHEMATICA
b[n_, i_, t_] := b[n, i, t] = If[n == 0, If[i == 0, {t!, 0}, {0, 0}], If[i < 1 || n < i*(i + 1)/2, {0, 0}, Sum[Function[p, p + {0, p[[1]]}*j][b[n - i*j, i - 1, t + j]/j!], {j, 1, n/i}]]];
a[n_] := Sum[b[n, k, 0][[2]], {k, 0, Floor[(Sqrt[1 + 8*n] - 1)/2]}];
Table[a[n], {n, 0, 32}] (* Jean-François Alcover, Jun 08 2024, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, May 31 2024
STATUS
approved