This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A138386 #14 Aug 27 2022 04:09:26 %S A138386 1,2,8,27,94,315,1067,3545,11767,38747,127061,414551,1347442,4362616, %T A138386 14078612,45290929,145291501,464864831,1483759703,4725204487, %U A138386 15016441266,47627848083,150784504858,476543143817,1503631824859 %N A138386 The initial values of the m-th row of table T of A137918 as m tends to infinity. %C A138386 It is clear that the first term of the m-th row of table T, a(0) = T(m, 3m) = T(1, 3) = 1. %C A138386 Let us call an x-partition of n a partition of n that has exactly x parts >= 3. %C A138386 Below we show that for j >= 1, a(j) = T(m, 3m + j) = T(j, 4j). %C A138386 An m-partition of n = 3m + j has at least m-j parts 3. Suppose that p is an m-partition of n with m-j-1 parts 3. The sum of the parts of p is at least n+1 since 3(m-j-1) + 4(m - (m-j-1)) = 3(m-j-1) + 4m -4(m-j-1) = 4m - (m-j-1) = 4m-m+j+1 = (3m + j) + 1. %C A138386 Because any m-partition of n = 3m + j has at least m-j parts 3, we obtain all the m-partitions of 3m + j if we append m-j parts 3 to each one of the j-partitions of 4j. Note that n - 3(m-j) = 4j and m - (m-j) = j. %C A138386 To complete we note that C(f(3) + K_3 - 1, K_3) = C(K_3, K_3) = 1 and the values taken by the product pi{3 <= i <= n}C(f(i) + K_i - 1, K_i) do not change when we add m-j to K_3. In fact pi{3 <= i <= n}C(f(i) + K_i - 1, K_i) = pi{4 <= i <= n}C(f(i) + K_i - 1, K_i). %C A138386 Using results found in the Knuth reference on pp. 9, ..., it is not difficult to show that the number of j-partitions of 4j is equal to p(j), (where p(n) is the usual partition function). %C A138386 The largest part that appears in those partitions is j+3, because j+4+3(j-1) = 4j+1. So the first 27 values of this sequence are showed. Note that f(k) is known for k <= 29 and we work with 1 <= j <= 26. %C A138386 With small values of 3m+j, the identity T(m, 3m+j) = T(j, 4j) comes closer to permit us to enumerate all the graphs with M components and N vertices, i.e., N >= 3, M >= 1. For example we determine T(M, 100) for M >= 25, T(M, 50) for M >= 8, T(M, 38) for M >= 4, T(M, 35) for M >= 3 and T(M, 32) for M >= 2. All graphs are determined if N <= 29. %C A138386 It can be shown that the formula computes T(i, 3i+j), i,j >= 1, for the nine values of i: floor((3i+j)/3) - 8 <= i <= floor((3i+j)/3). %H A138386 D. E. Knuth, <a href="http://www-cs-faculty.stanford.edu/~knuth/fasc3b.ps.gz">The Art of Computer Programming</a>, Vol. 4, Pre-Fascicle 3b, see p. 9. %F A138386 a(0) = 1; for j >= 1, a(j) = Sum over the partitions 3K_3 + ... + nK_n of n = 4j with exactly j parts of pi{4 <= i <= n} C(f(i) + K_i - 1, K_i), where f(i) is A001429(i). %F A138386 Euler transform of 2, 5, 13, 33, ... (A001429). - _Vladeta Jovovic_, Sep 17 2008 %e A138386 Choosing j = 5 we use p(5) or 7 partitions. The largest part appearing in those partitions is 8, so we need the first 6 values of A001429, given by %e A138386 ...i..|3.4..5...6...7...8 %e A138386 ------+------------------ %e A138386 f(i)..|1.2..5..13..33..89 %e A138386 Using the program GAP, the partitions produced by the command "RestrictedPartitions(20, [3..8], 5);" are: %e A138386 [ [ 4, 4, 4, 4, 4 ], [ 5, 4, 4, 4, 3 ], [ 5, 5, 4, 3, 3 ], [ 6, 4, 4, 3, 3 ], [ 6, 5, 3, 3, 3 ], [ 7, 4, 3, 3, 3 ], [ 8, 3, 3, 3, 3 ] ] %e A138386 Eliminating parts 3 from those partitions, below we give them in the form 4K_4 +...+ nK_n followed by the corresponding number of graphs: %e A138386 4*5 -> C(2+5-1, 5) = 6 %e A138386 5 + 4*3 -> 5C(2+3-1, 3) = 20 %e A138386 5*2 + 4 -> 2C(5+2-1, 2) = 30 %e A138386 6 + 4*2 -> 13C(2+2-1, 2) = 39 %e A138386 6 + 5 -> 13 * 5 = 65 %e A138386 7 + 4 -> 33 * 2 = 66 %e A138386 8 -> 89 %e A138386 So a(5) = 6 + 20 + 30 + 39 + 65 + 66 + 89 = 315. %e A138386 T(333332, 10^6+1) = a(5) = 315. Note that j = 5 and m = 333332 gives 3m+j = 10^6+1. %Y A138386 Cf. A137918, A001429. %K A138386 nonn %O A138386 0,2 %A A138386 _Washington Bomfim_, Mar 18 2008