Search: id:A093694
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%I A093694
%S A093694 1,2,5,9,17,27,46,69,108,158,234,331,476,657,915,1244,1694,2262,3029,
%T A093694 3988,5257,6844,8901,11461,14749,18809,23958,30304,38263,48018,60167,
%U A093694 74977,93276,115509,142772,175759,215991,264449,323216,393772,478884
%N A093694 Number of one-element transitions from the partitions of n to the partitions 
               of n+1 for labeled parts.
%C A093694 For the unlabeled case, the number of one-element transitions from the 
               partitions of n to the partitions of n+1 is given by A000070. Example: 
               There are A000070(4) = 12 transitions from n=4 to n=5: [1111] -> 
               [11111], [1111] -> [1112], [112] -> [1112], [112] -> 113], [112] 
               -> [122], [13] -> [113], [13] -> [14], [13] -> [23], [22] -> [23], 
               [22] -> [122], [4] -> [14], [4] -> [5].
%C A093694 a(n) is also the total number of digits in all partitions of the integer 
               n which contain at least one digit 1.
%H A093694 T. Wieder, <a href="http://www.thomas-wieder.privat.t-online.de">Home 
               Page</a>.
%H A093694 T. Wieder, <a href="http://homepages.tu-darmstadt.de/~wieder">(old) Home 
               Page</a>.
%F A093694 a(n) = Sum_k=1^p(n) (nops(p(k, n)) + 1), where p(n) is the number of 
               partitions of n and nops(p(k, n)) is the number of parts in the k-th 
               partition p(n, k) of n.
%F A093694 a(n) = Sum_k=1^p(n) nops(p(k, n)[subject to: at least one p(l, k, n) 
               = 1]; p(n) = number of partitions of n, p(k, n) = k-th partition, 
               p(l, k, n) = l-th part in the k-th partition p(k, n) of integer n.
%e A093694 In the labeled case, we have 9 one-element transitions from all partitions 
               of n=3 to the partitions of n+1=4: [1,1,1] -> [1,1,1,1]; [1,1,1] 
               -> [1,1,2]; [1,1,1] -> [1,1,2]; [1,1,1] -> [1,1,2]; [1,2] -> [1,1,
               2]; [1,2] -> [1,3]; [1,2] -> [2,2]; [3] -> [1,3]; [3] -> [4].
%e A093694 For n=4 we have the following partitions which contain at least one digit 
               1: [1111], [112], [13] and these partitions contain 4 + 3 + 2 = 9 
               = a(4) digits.
%p A093694 main := proc(n::integer) local a,ndxp,ListOfPartitions; with(combinat): 
               with(ListTools): ListOfPartitions:=partition(n-1); a:=0; for ndxp 
               from 1 to nops(ListOfPartitions) do if Occurrences(1, ListOfPartitions[ndxp]) 
               > 0 then a:=a+nops(Flatten(ListOfPartitions[ndxp])); print("ndxp, 
               Flatten(ListOfPartitions[ndxp]):",ndxp, Flatten(ListOfPartitions[ndxp])); 
               print("ndxp, ListOfPartitions[ndxp], a:",ndxp, ListOfPartitions[ndxp],
               a); # End of if-clause *** Occurrences(1, ListOfPartitions[ndxp]) 
               *** fi; end do; print("n, a(n):",n,a); end proc;
%t A093694 (* first do *) Needs["DiscreteMath`Combinatorica`"] (* then *) f[n_] 
               := Block[{l = Sort[ Flatten[ Partitions[n]]]}, Length[l] - Count[l, 
               1]]; g[n_] := (f[n] + Sum[PartitionsP[k], {k, 0, n}]); Table[ g[n], 
               {n, 0, 40}] (from Robert G. Wilson v Jul 13 2004)
%Y A093694 Cf. A000070, A093695, A089378.
%Y A093694 Sequence in context: A062492 A165271 A139672 this_sequence A068006 A000097 
               A081996
%Y A093694 Adjacent sequences: A093691 A093692 A093693 this_sequence A093695 A093696 
               A093697
%K A093694 nonn
%O A093694 0,2
%A A093694 Thomas Wieder (wieder.thomas(AT)t-online.de), Apr 10 2004
%E A093694 More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Jul 13 2004

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