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A335941 Number of partitions of n such that the set s of parts and multiplicities satisfies s = {1..max(s)}.

%I A335941 #21 May 30 2022 08:11:34
%S A335941 1,1,2,1,1,4,2,5,5,9,8,15,11,14,22,28,30,36,37,53,60,80,83,104,114,
%T A335941 148,157,201,218,283,284,362,400,455,518,624,697,807,907,1036,1181,
%U A335941 1368,1531,1727,1990,2197,2563,2849,3182,3568,4095,4548,5143,5720,6420
%N A335941 Number of partitions of n such that the set s of parts and multiplicities satisfies s = {1..max(s)}.
%H A335941 Chai Wah Wu, <a href="/A335941/b335941.txt">Table of n, a(n) for n = 0..158</a> (n = 0..120 from Alois P. Heinz)
%e A335941 a(0) = 1: the empty partition.
%e A335941 a(1) = 1: 1.
%e A335941 a(2) = 2: 11, 2.
%e A335941 a(3) = 1: 21.
%e A335941 a(4) = 1: 211.
%e A335941 a(5) = 4: 2111, 221, 311, 32.
%e A335941 a(6) = 2: 2211, 321.
%e A335941 a(7) = 5: 22111, 2221, 3211, 322, 331.
%e A335941 a(8) = 5: 22211, 32111, 3221, 3311, 332.
%e A335941 a(9) = 9: 222111, 321111, 32211, 3222, 33111, 3321, 42111, 4311, 432.
%e A335941 a(10) = 8: 2221111, 322111, 32221, 331111, 33211, 4222, 4321, 433.
%p A335941 b:= proc(n,i,s) option remember;
%p A335941      `if`(n=0, `if`(s={$0..max(s)}, 1, 0), `if`(i<1, 0, add(
%p A335941         b(n-i*j, i-1, {s[], j, `if`(j=0, 0, i)}), j=0..n/i)))
%p A335941     end:
%p A335941 a:= n-> b(n, floor((sqrt(1+8*(n+1))-1)/2), {0}):
%p A335941 seq(a(n), n=0..55);
%t A335941 b[n_, i_, s_] := b[n, i, s] =
%t A335941      If[n == 0, If[s == Range[0, Max[s]], 1, 0], If[i < 1, 0, Sum[
%t A335941      b[n-i*j, i-1, Union@Flatten@{s, j, If[j == 0, 0, i]}], {j, 0, n/i}]]];
%t A335941 a[n_] := b[n, Floor[(Sqrt[1 + 8*(n + 1)] - 1)/2], {0}];
%t A335941 Table[a[n], {n, 0, 55}] (* _Jean-François Alcover_, May 30 2022, after _Alois P. Heinz_ *)
%Y A335941 Cf. A317081, A317088, A335942.
%K A335941 nonn
%O A335941 0,3
%A A335941 _Alois P. Heinz_, Jun 30 2020