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A336718 Total number of left-to-right maxima in all compositions of n into distinct parts. 4
0, 1, 1, 4, 4, 7, 18, 21, 32, 46, 107, 121, 193, 257, 379, 728, 900, 1299, 1806, 2529, 3360, 6182, 7387, 10807, 14385, 20217, 26207, 36450, 58194, 72887, 101130, 135379, 183178, 240796, 323307, 417625, 649959, 797623, 1096645, 1426108, 1931340, 2470541 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
Also total number of left-to-right minima in all compositions of n into distinct parts.
LINKS
FORMULA
a(n) = Sum_{k=1..floor((sqrt(8*n+1)-1)/2)} A000254(k) * A008289(n,k).
EXAMPLE
a(6) = 18 = 3+2+2+2+1+1+2+1+2+1+1: (1)(2)(3), (1)(3)2, (2)1(3), (2)(3)1, (3)12, (3)21, (2)(4), (4)2, (1)(5), (5)1, (6).
MAPLE
g:= proc(n) option remember;
`if`(n<2, n, (2*n-1)*g(n-1)-(n-1)^2*g(n-2))
end:
b:= proc(n, i, p) option remember; `if`(i*(i+1)/2<n, 0,
`if`(n=0, g(p), b(n, i-1, p)+b(n-i, min(n-i, i-1), p+1)))
end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..50);
MATHEMATICA
g[n_] := g[n] = If[n < 2, n, (2 n - 1) g[n - 1] - (n - 1)^2 g[n - 2]];
b[n_, i_, p_] := b[n, i, p] = If[i (i + 1)/2 < n, 0, If[n == 0, g[p], b[n, i - 1, p] + b[n - i, Min[n - i, i - 1], p + 1]]];
a[n_] := b[n, n, 0];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jul 12 2021, after Alois P. Heinz *)
CROSSREFS
Sequence in context: A115292 A202676 A330765 * A173324 A318243 A321661
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Aug 01 2020
STATUS
approved

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Last modified March 29 08:59 EDT 2024. Contains 371268 sequences. (Running on oeis4.)