A050315 - OEIS

%I #25 May 21 2021 08:11:19

%S 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,

%T 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,

%U 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

%N Main diagonal of A050314.

%C 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

%C From _Gus Wiseman_, Mar 30 2019: (Start)

%C 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:

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

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

%C                            (52)                (92)       (94)  (A4)  (96)

%C                            (61)                (A1)       (C1)  (C2)  (A5)

%C                            (421)               (821)      (841) (842) (B4)

%C                                                                       (C3)

%C                                                                       (D2)

%C                                                                       (E1)

%C                                                                       (843)

%C                                                                       (852)

%C                                                                       (861)

%C                                                                       (942)

%C                                                                       (A41)

%C                                                                       (C21)

%C                                                                       (8421)

%C (End)

%H Alois P. Heinz, <a href="/A050315/b050315.txt">Table of n, a(n) for n = 0..16383</a>

%H Michael Gilleland, <a href="/selfsimilar.html">Some Self-Similar Integer Sequences</a>

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

%p a:= n-> combinat[bell](add(i,i=convert(n, base, 2))):

%p seq(a(n), n=0..100);  # _Alois P. Heinz_, Apr 08 2019

%t binpos[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];

%t stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];

%t Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&stableQ[#,Intersection[binpos[#1],binpos[#2]]!={}&]&]],{n,0,20}] (* _Gus Wiseman_, Mar 30 2019 *)

%t a[n_] := BellB[DigitCount[n, 2, 1]];

%t a /@ Range[0, 100] (* _Jean-François Alcover_, May 21 2021 *)

%Y Cf. A000110, A000120, A050314.

%Y Cf. A070939, A080572, A247935, A267610.

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

%Y Main diagonal of A307431 and of A307505.

%K nonn

%O 0,4

%A _Christian G. Bower_, Sep 15 1999