A050315 Main diagonal of A050314.

1, 1, 1, 2, 1, 2, 2, 5, 1, 2, 2, 5, 2, 5, 5, 15, 1, 2, 2, 5, 2, 5, 5, 15, 2, 5, 5, 15, 5, 15, 15, 52, 1, 2, 2, 5, 2, 5, 5, 15, 2, 5, 5, 15, 5, 15, 15, 52, 2, 5, 5, 15, 5, 15, 15, 52, 5, 15, 15, 52, 15, 52, 52, 203, 1, 2, 2, 5, 2, 5, 5, 15, 2, 5, 5, 15, 5, 15, 15, 52, 2, 5, 5, 15, 5, 15

Offset: 0

Author

Christian G. Bower, Sep 15 1999

Keywords

• nonn

Comments

Also, a(n) is the number of odd multinomial coefficients n!/(k_1!...k_m!) with 1 <= k_1 <= ... <= k_m and k_1 + ... + k_m = n. - Pontus von Brömssen, Mar 23 2018

From Gus Wiseman, Mar 30 2019: (Start)

Also the number of strict integer partitions of n with no binary carries. The Heinz numbers of these partitions are given by A325100. A binary carry of two positive integers is an overlap of the positions of 1's in their reversed binary expansion. For example, the a(1) = 1 through a(15) = 15 strict integer partitions with no binary carries are:

(1) (2) (3) (4) (5) (6) (7) (8) (9) (A) (B) (C) (D) (E) (F)

(21) (41) (42) (43) (81) (82) (83) (84) (85) (86) (87)

(52) (92) (94) (A4) (96)

(61) (A1) (C1) (C2) (A5)

(421) (821) (841) (842) (B4)

(C3)

(D2)

(E1)

(843)

(852)

(861)

(942)

(A41)

(C21)

(8421)

(End)

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..16383
• Michael Gilleland, Some Self-Similar Integer Sequences

Crossrefs

Cf. A000110, A000120, A050314.

Cf. A070939, A080572, A247935, A267610.

Cf. A325093, A325095, A325096, A325099, A325100, A325103, A325110, A325123.

Main diagonal of A307431 and of A307505.

Programs

Maple

a:= n-> combinat[bell](add(i,i=convert(n, base, 2))):
seq(a(n), n=0..100);  # Alois P. Heinz, Apr 08 2019

Mathematica

binpos[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&stableQ[#,Intersection[binpos[#1],binpos[#2]]!={}&]&]],{n,0,20}] (* Gus Wiseman, Mar 30 2019 *)
a[n_] := BellB[DigitCount[n, 2, 1]];
a /@ Range[0, 100] (* Jean-François Alcover, May 21 2021 *)

Formula

Bell number of number of 1's in binary: a(n) = A000110(A000120(n)).