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%I M0283 N0101 #68 Oct 22 2017 22:08:25
%S 1,1,1,1,2,2,3,4,5,7,9,11,15,18,23,30,37,47,58,71,90,110,136,164,201,
%T 248,300,364,436,525,638,764,919,1090,1297,1549,1845,2194,2592,3060,
%U 3590,4242,5013,5888,6912,8070,9418,11004,12866,15021,17475,20298,23501,27169
%N Max_{k=0..n} { Number of partitions of n into exactly k parts }.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Robert Israel, <a href="/A002569/b002569.txt">Table of n, a(n) for n = 0..3000</a> (n = 0..97 from Robert G. Wilson v)
%H F. C. Auluck, S. Chowla and H. Gupta, <a href="/A002569/a002569.pdf">On the maximum value of the number of partitions into k parts</a>, J. Indian Math. Soc., 6 (1942), 105-112. [Annotated scanned copy. But the last page is in a separate file: see the next link.]
%H F. C. Auluck, S. Chowla and H. Gupta, <a href="/A002569/a002569_1.pdf">On the maximum value of the number of partitions into k parts</a>, [Annotated scanned copy of page 112 only]
%H Victor Meally, <a href="/A002868/a002868.pdf">Comparison of several sequences given in Motzkin's paper "Sorting numbers for cylinders...", letter to N. J. A. Sloane, N. D.</a>
%H T. S. Motzkin, <a href="/A000262/a000262.pdf">Sorting numbers for cylinders and other classification numbers</a>, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176 (p. 172, gives a(9) incorrectly as 6). [Annotated, scanned copy]
%H OEIS Wiki, <a href="http://oeis.org/wiki/Sorting_numbers">Sorting numbers</a>
%p 1, seq(max(seq(combinat:-numbpart(n,k)-combinat:-numbpart(n,k-1),k=1..n)),n=1..100); # _Robert Israel_, Nov 24 2014
%t f[n_] := Block[{k = 1, mx = 0}, While[k < n + 1, a = Length@ IntegerPartitions[n, {k}]; If[a > mx, mx = a]; k++ ]; mx]; Array[f, 53] (* _Robert G. Wilson v_, Jul 20 2010 *)
%t t[0, k_] := 1; t[1, k_] := 1 /; k > 0; t[n_, k_] := 0 /; n < 0; t[n_, 0] := 0 /; n > 0; t[n_, 1] := 1 /; n > 0; t[n_, k_] := t[n, k] = Sum[t[n - k + i, k - i], {i, 0, k - 1}];
%t f[n_] := Max[ Table[ t[n - k, k], {k, 0, n}]]; Array[f, 54, 0] (* _Robert G. Wilson v_, Nov 24 2014 *)
%t Max[CoefficientList[#, a]] & /@ (1/QPochhammer[a q, q] + O[q]^60)[[3]] (* _Vladimir Reshetnikov_, Nov 17 2016 *)
%Y Cf. A000041, A008284, A026819, A046155.
%K nonn
%O 0,5
%A _N. J. A. Sloane_
%E More terms from _David W. Wilson_
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