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%I #83 Aug 16 2015 09:07:09
%S 1,4,13,35,86,194,415,844,1654,3133,5773,10372,18240,31449,53292,
%T 88873,146095,236977,379746,601656,943305,1464501,2252961,3436182,
%U 5198644,7805248,11634685,17224795,25336141,37038139,53828275,77792869
%N Level of the n-th plateau of the column k of the square array A195825, when k -> infinity.
%C 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 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 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 Also numbers that are repeated in column k of square array A195825, when k -> infinity.
%C 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 Is this the EULER transform of 4, 3, 3, 3, 3, 3, 3...?
%F From _Vaclav Kotesovec_, Aug 16 2015: (Start)
%F a(n) ~ sqrt(2*n)/Pi * A000716(n).
%F a(n) ~ exp(sqrt(2*n)*Pi) / (8*Pi*n).
%F (End)
%e 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 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 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 CoefficientList[Series[1/(1-x)*Product[1/(1-x^k)^3,{k,1,50}],{x,0,50}],x] (* _Vaclav Kotesovec_, Aug 16 2015 *)
%o (GWBASIC).
%o 10 'This program gives the 32 terms of DATA section.
%o 20 'Suppose that we have A057077().
%o 30 'In this case g(n) is the n-th generalized 64-gonal number.
%o 40 DEFDBL a, g, w
%o 50 DIM a(32), A057077(2079), g(2080), w(2079)
%o 60 n=0: w(0)=1
%o 70 FOR i = 1 TO 2079
%o 80 FOR j = 1 TO i
%o 90 IF g(j)<=i THEN w(i)=w(i)+A057077(j-1)*w(i-g(j))
%o 100 NEXT j
%o 110 IF i=1 GOTO 130
%o 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 130 NEXT i
%o 140 END
%Y Partial sums of A000716. Column 3 of A210764.
%Y Columns (k=1..10) of A195825: A000041, A006950, A036820, A195848, A195849, A195850, A195851, A195852, A196933, A210964.
%Y Cf. A000070, A000712, A000713, A010815, A211970.
%K nonn
%O 1,2
%A _Omar E. Pol_, Jun 19 2012
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