cp's OEIS Frontend

A387179 Number of twice-partitions of n into distinct constant partitions.

%I A387179 #12 Sep 09 2025 18:26:33
%S A387179 1,1,2,4,7,11,18,28,48,69,105,158,240,343,503,720,1041,1459,2062,2874,
%T A387179 4047,5547,7656,10472,14322,19360,26214,35192,47354,63030,83992,
%U A387179 111258,147360,193804,254907,333553,436319,567673,738197,956049,1237453,1594700,2053361
%N A387179 Number of twice-partitions of n into distinct constant partitions.
%C A387179 A twice-partition of n (A063834) is a sequence of integer partitions, one of each part of an integer partition of n.
%H A387179 Alois P. Heinz, <a href="/A387179/b387179.txt">Table of n, a(n) for n = 0..5000</a>
%e A387179 The a(1) = 1 through a(6) = 18 twice-partitions counted by this sequence:
%e A387179   (1)  (2)   (3)      (4)       (5)         (6)
%e A387179        (11)  (111)    (22)      (3)(2)      (33)
%e A387179              (2)(1)   (1111)    (4)(1)      (222)
%e A387179              (11)(1)  (3)(1)    (11111)     (4)(2)
%e A387179                       (11)(2)   (22)(1)     (5)(1)
%e A387179                       (2)(11)   (3)(11)     (22)(2)
%e A387179                       (111)(1)  (111)(2)    (4)(11)
%e A387179                                 (111)(11)   (111111)
%e A387179                                 (1111)(1)   (111)(3)
%e A387179                                 (11)(2)(1)  (22)(11)
%e A387179                                 (2)(11)(1)  (3)(111)
%e A387179                                             (1111)(2)
%e A387179                                             (3)(2)(1)
%e A387179                                             (1111)(11)
%e A387179                                             (11111)(1)
%e A387179                                             (3)(11)(1)
%e A387179                                             (111)(2)(1)
%e A387179                                             (111)(11)(1)
%p A387179 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(j!*
%p A387179       binomial(numtheory[tau](i), j)*b(n-i*j, i-1), j=0..n/i)))
%p A387179     end:
%p A387179 a:= n-> b(n$2):
%p A387179 seq(a(n), n=0..45);  # _Alois P. Heinz_, Sep 08 2025
%t A387179 twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn],{ptn,IntegerPartitions[n]}];
%t A387179 Table[Length[Select[twiptn[n],UnsameQ@@#&&And@@SameQ@@@#&]],{n,0,10}]
%Y A387179 The non-distinct version is A279784.
%Y A387179 Dominates the case of distinct block-sums A279786.
%Y A387179 This is the constant-block case of A296122.
%Y A387179 For strict instead of constant partitions we have A358914.
%Y A387179 A000041 counts integer partitions, strict A000009.
%Y A387179 A047968 counts constant twice-partitions.
%Y A387179 A063834 counts twice-partitions.
%Y A387179 Cf. A387120, zeros A387180 (counted by A387329), nonzeros A387181 (counted by A387330).
%Y A387179 Cf. A000005, A006906, A007425, A261049, A270995, A299200, A299201, A355739, A387110.
%K A387179 nonn,new
%O A387179 0,3
%A A387179 _Gus Wiseman_, Sep 07 2025
%E A387179 More terms from _Alois P. Heinz_, Sep 08 2025