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A210843 Level of the n-th plateau of the column k of the square array A195825, when k -> infinity.

%I A210843 #86 May 17 2024 19:47:06
%S A210843 1,4,13,35,86,194,415,844,1654,3133,5773,10372,18240,31449,53292,
%T A210843 88873,146095,236977,379746,601656,943305,1464501,2252961,3436182,
%U A210843 5198644,7805248,11634685,17224795,25336141,37038139,53828275,77792869
%N A210843 Level of the n-th plateau of the column k of the square array A195825, when k -> infinity.
%C A210843 Also the first (k+1)/2 terms of this sequence are the levels of the (k+1)/2 plateaus of the column k of A195825, whose lengths are k+1, k-1, k-3, k-5,... 2, if k is odd.
%C A210843 Also the first k/2 terms of this sequence are the levels of the k/2 plateaus of the column k of A195825, whose lengths are k+1, k-1, k-3, k-5,... 3, if k is a positive even number.
%C A210843 For the visualization of the plateaus see the graph of the sequences mentioned in crossrefs section (columns k=1..10 of A195825), for example see the graph of A210964.
%C A210843 Also numbers that are repeated in column k of square array A195825, when k -> infinity.
%C A210843 Note that the definition and the comments related to the square array A195825 mentioned above are also valid for the square array A211970, since both arrays contains the same columns, if k >= 1.
%C A210843 Is this the EULER transform of 4, 3, 3, 3, 3, 3, 3...?
%F A210843 From _Vaclav Kotesovec_, Aug 16 2015: (Start)
%F A210843 a(n) ~ sqrt(2*n)/Pi * A000716(n).
%F A210843 a(n) ~ exp(sqrt(2*n)*Pi) / (8*Pi*n).
%F A210843 (End)
%e A210843 Column 1 of A195825 is A000041 which starts: [1, 1], 2, 3, 5, 7, 11... The column contains only one plateau: [1, 1] which has level 1 and length 2. So a(1) = 1.
%e A210843 Column 3 of A195825 is A036820 which starts: [1, 1, 1, 1], 2, 3, [4, 4], 5, 7, 10... The column contains only two plateaus: [1, 1, 1, 1], [4, 4], which have levels 1, 4 and lengths 4, 2. So a(1)= 1 and a(2) = 2.
%e A210843 Column 6 of A195825 is A195850 which starts: [1, 1, 1, 1, 1, 1, 1], 2, 3, [4, 4, 4, 4, 4], 5, 7, 10, 12, [13, 13, 13], 14, 16, 21... The column contains three plateaus: [1, 1, 1, 1, 1, 1, 1], [4, 4, 4, 4, 4], [13, 13, 13], which have levels 1, 4, 13 and lengths 7, 5, 3. So a(1) = 1, a(2) = 4 and a(3) = 13.
%t A210843 CoefficientList[Series[1/(1-x)*Product[1/(1-x^k)^3,{k,1,50}],{x,0,50}],x] (* _Vaclav Kotesovec_, Aug 16 2015 *)
%o A210843 (GW-BASIC)
%o A210843 10 'This program gives the 32 terms of DATA section.
%o A210843 20 'Suppose that we have A057077().
%o A210843 30 'In this case g(n) is the n-th generalized 64-gonal number.
%o A210843 40 DEFDBL a, g, w
%o A210843 50 DIM a(32), A057077(2079), g(2080), w(2079)
%o A210843 60 n=0: w(0)=1
%o A210843 70 FOR i = 1 TO 2079
%o A210843 80   FOR j = 1 TO i
%o A210843 90     IF g(j)<=i THEN w(i)=w(i)+A057077(j-1)*w(i-g(j))
%o A210843 100   NEXT j
%o A210843 110  IF i=1 GOTO 130
%o A210843 120  IF w(i-2)=w(i-1) AND w(i-1)<>a(n) THEN n=n+1: a(n)=w(i-1): PRINT a(n);
%o A210843 130 NEXT i
%o A210843 140 END
%Y A210843 Partial sums of A000716. Column 3 of A210764.
%Y A210843 Columns (k=1..10) of A195825: A000041, A006950, A036820, A195848, A195849, A195850, A195851, A195852, A196933, A210964.
%Y A210843 Cf. A000070, A000712, A000713, A010815, A211970.
%K A210843 nonn
%O A210843 1,2
%A A210843 _Omar E. Pol_, Jun 19 2012