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A323952 Regular triangle read by rows where if k > 1 then T(n, k) is the number of connected subgraphs of an n-cycle with any number of vertices other than 2 through k - 1, n >= 1, 1 <= k <= n - 1. Otherwise T(n, 1) = n.

%I A323952 #12 Jan 19 2023 16:36:48
%S A323952 1,2,3,3,7,4,4,13,9,5,5,21,16,11,6,6,31,25,19,13,7,7,43,36,29,22,15,8,
%T A323952 8,57,49,41,33,25,17,9,9,73,64,55,46,37,28,19,10,10,91,81,71,61,51,41,
%U A323952 31,21,11,11,111,100,89,78,67,56,45,34,23,12,12,133,121
%N A323952 Regular triangle read by rows where if k > 1 then T(n, k) is the number of connected subgraphs of an n-cycle with any number of vertices other than 2 through k - 1, n >= 1, 1 <= k <= n - 1. Otherwise T(n, 1) = n.
%H A323952 Andrew Howroyd, <a href="/A323952/b323952.txt">Table of n, a(n) for n = 1..1275</a> (rows 1..50)
%F A323952 T(n, 1) = n; T(n, k) = 1 + n * (n - k + 1).
%e A323952 Triangle begins:
%e A323952    1
%e A323952    2   3
%e A323952    3   7   4
%e A323952    4  13   9   5
%e A323952    5  21  16  11   6
%e A323952    6  31  25  19  13   7
%e A323952    7  43  36  29  22  15   8
%e A323952    8  57  49  41  33  25  17   9
%e A323952    9  73  64  55  46  37  28  19  10
%e A323952   10  91  81  71  61  51  41  31  21  11
%e A323952   11 111 100  89  78  67  56  45  34  23  12
%e A323952   12 133 121 109  97  85  73  61  49  37  25  13
%e A323952 Row 4 counts the following connected sets:
%e A323952   {1}  {1}     {1}     {1}
%e A323952   {2}  {2}     {2}     {2}
%e A323952   {3}  {3}     {3}     {3}
%e A323952   {4}  {4}     {4}     {4}
%e A323952        {12}    {123}   {1234}
%e A323952        {14}    {124}
%e A323952        {23}    {134}
%e A323952        {34}    {234}
%e A323952        {123}   {1234}
%e A323952        {124}
%e A323952        {134}
%e A323952        {234}
%e A323952        {1234}
%t A323952 anesw[n_,k_]:=Length[If[k==1,List/@Range[n],Union[Sort/@Select[Union[List/@Range[n],Join@@Table[Partition[Range[n],i,1,1],{i,k,n}]],UnsameQ@@#&&#!={}&]]]];
%t A323952 Table[anesw[n,k],{n,0,16},{k,n}]
%o A323952 (PARI) T(n,k) = if(k==1, n, 1 + n * (n - k + 1)) \\ _Andrew Howroyd_, Jan 18 2023
%Y A323952 First column is A000027. Second column is A002061. Third column is A000290. Fourth column is A028387.
%Y A323952 Cf. A000126, A000325, A306351, A323950, A323951, A323953, A323954, A323956.
%K A323952 nonn,tabl
%O A323952 1,2
%A A323952 _Gus Wiseman_, Feb 10 2019