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A365903
Number of partitions of [n] whose block minima sum to k, where k is chosen so as to maximize this number.
3
1, 1, 1, 2, 4, 10, 29, 101, 367, 1562, 6891, 37871, 197930, 1121634, 6888085, 46190282, 323250987, 2349020516, 17897285514, 142512956148, 1178963284732, 10248806222398, 91421283039658, 847666112839362, 8100455404172267, 79925567946537362, 814508927747776069
OFFSET
0,4
LINKS
MAPLE
b:= proc(n, i, t, m) option remember; `if`(n=0, t^(m-i+1),
`if`((i+m)*(m+1-i)/2<n or i>n, 0, `if`(t=0, 0,
t*b(n, i+1, t, m))+ b(n-i, i+1, t+1, m)))
end:
a:= n-> max(seq(b(k, 1, 0, n), k=0..n*(n+1)/2)):
seq(a(n), n=0..26);
# second Maple program:
a:= proc(h) option remember; local b; b:=
proc(n, m) option remember; `if`(n=0, 1,
b(n-1, m)*m + expand(x^(h-n+1)*b(n-1, m+1)))
end: forget(b); max(coeffs(b(h, 0)))
end:
seq(a(n), n=0..26);
MATHEMATICA
Q[1, t_, s_] := t*s;
Q[n_, t_, s_] := Q[n, t, s] = s*D[Q[n-1, t, s], s] + s*t^n*Q[n-1, t, s] // Expand;
P[n_, t_] := Module[{s}, Q[n, t, s] /. s -> 1];
a[n_] := If[n == 0, 1, Module[{t}, CoefficientList[P[n, t], t] // Max]];
Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Oct 03 2024 *)
CROSSREFS
Row maxima of A124327.
Cf. A367969.
Sequence in context: A261041 A047051 A271077 * A126349 A076315 A304124
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Dec 14 2023
STATUS
approved