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A050315
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Main diagonal of A050314.
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23
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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
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OFFSET
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0,4
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COMMENTS
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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
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)
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LINKS
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FORMULA
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MAPLE
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a:= n-> combinat[bell](add(i, i=convert(n, base, 2))):
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MATHEMATICA
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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]];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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