A309670 Number of colored compositions of n using all colors of an initial interval of the color palette such that all parts have different color patterns and the patterns for parts i are sorted and have i colors in (weakly) increasing order.
1, 1, 3, 21, 115, 813, 7627, 71173, 740023, 8544169, 107195083, 1434581205, 20499413667, 312262663989, 4992164670007, 84221279919193, 1492818584618099, 27607818180267269, 533522844488072987, 10724970103003953053, 223859943086157531063, 4847766598150865273721
Offset: 0
Keywords
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..200
Crossrefs
Row sums of A327244.
Programs
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Maple
C:= binomial: b:= proc(n, i, k, p) option remember; `if`(n=0, p!, `if`(i<1, 0, add( b(n-i*j, min(n-i*j, i-1), k, p+j)/j!*C(C(k+i-1, i), j), j=0..n/i))) end: a:= n-> add(add(b(n$2, i, 0)*(-1)^(k-i)*C(k, i), i=0..k), k=0..n): seq(a(n), n=0..23);
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Mathematica
c = Binomial; b[n_, i_, k_, p_] := b[n, i, k, p] = If[n == 0, p!, If[i<1, 0, Sum[b[n - i*j, Min[n - i*j, i-1], k, p+j]/j!*c[c[k+i-1, i], j], {j, 0, n/i}]]]; a[n_] := Sum[Sum[b[n, n, i, 0]*(-1)^(k-i)*c[k, i], {i, 0, k}], {k, 0, n}]; Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Mar 05 2022, after Alois P. Heinz *)