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A335942 Number of compositions of n such that the set s of parts and multiplicities satisfies s = {1..max(s)}.

%I A335942 #18 May 30 2022 08:12:36
%S A335942 1,1,2,2,3,12,12,32,51,144,191,486,679,1487,3149,5909,11637,18630,
%T A335942 36928,76431,141009,264784,535057,921105,1774022,3388054,6303519,
%U A335942 12255373,22527578,43358822,77695383,145170435,264722429,527776034,936538336,1807344134
%N A335942 Number of compositions of n such that the set s of parts and multiplicities satisfies s = {1..max(s)}.
%H A335942 Chai Wah Wu, <a href="/A335942/b335942.txt">Table of n, a(n) for n = 0..142</a> (n = 0..100 from Alois P. Heinz)
%e A335942 a(4) = 3: 211, 121, 112.
%e A335942 a(5) = 12: 23, 32, 113, 122, 131, 212, 221, 311, 1112, 1121, 1211, 2111.
%p A335942 b:= proc(n, i, s, p) option remember;
%p A335942      `if`(n=0, `if`(s={$0..max(s)}, p!, 0), `if`(i<1, 0, add(
%p A335942       b(n-i*j, i-1, {s[], j, `if`(j=0, 0, i)}, p+j)/j!, j=0..n/i)))
%p A335942     end:
%p A335942 a:= n-> b(n, floor((sqrt(1+8*(n+1))-1)/2), {0}, 0):
%p A335942 seq(a(n), n=0..35);
%t A335942 b[n_, i_, s_, p_] := b[n, i, s, p] =
%t A335942      If[n == 0, If[s == Range[0, Max[s]], p!, 0], If[i < 1, 0, Sum[
%t A335942      b[n - i*j, i - 1, Union@Flatten@{s, j, If[j == 0, 0, i]}, p + j]/j!,
%t A335942      {j, 0, n/i}]]];
%t A335942 a[n_] := b[n, Floor[(Sqrt[1 + 8*(n + 1)] - 1)/2], {0}, 0];
%t A335942 Table[a[n], {n, 0, 35}] (* _Jean-François Alcover_, May 30 2022, after _Alois P. Heinz_ *)
%Y A335942 Cf. A107429, A329741, A335941.
%K A335942 nonn
%O A335942 0,3
%A A335942 _Alois P. Heinz_, Jun 30 2020