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 A047993 Number of balanced partitions of n: the largest part equals the number of parts. 50
 1, 0, 1, 1, 1, 1, 3, 2, 4, 4, 6, 7, 11, 11, 16, 19, 25, 29, 40, 45, 60, 70, 89, 105, 134, 156, 196, 232, 285, 336, 414, 485, 591, 696, 839, 987, 1187, 1389, 1661, 1946, 2311, 2702, 3201, 3731, 4400, 5126, 6018, 6997, 8195, 9502, 11093, 12849, 14949, 17281, 20062 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,7 COMMENTS Useful in the creation of plane partitions with C3 or C3v symmetry. The function T[m,a,b] used here gives the partitions of m whose Ferrers plot fits within an a X b box. Central terms of triangle in A063995: a(n) = A063995(n,0). - Reinhard Zumkeller, Jul 24 2013 Sequence enumerates the collection of partitions of size n that are in the semigroup of Dyson rank=0, or balanced partitions, under the binary operation A*B = (a1,a2,...,a[k-1],k)*(b1,...,b[n-1,n) = (a1*b1,...,a1*n,a2*b1,...,a2*n,...,k*b1,...,k*n), where A is a partition with k parts and B is a partition with n parts, and A*B is a partition with k*n parts. Note that the rank of A*B is 0, as required. For example, the product of the rank 0 partitions (1,2,3) of 6 and (1,1,3) of 5 is the rank 0 partition (1,1,2,2,3,3,3,6,9) of 30. There is no rank zero partition of 2, as shown in the sequence. It can be seen that any element of the semigroup that partitions an odd prime p or a composite number of form 2p cannot be a product of smaller nontrivial partitions, whether in this semigroup or not. - Richard Locke Peterson, Jul 15 2018 The Heinz numbers of these integer partitions are given by A106529. - Gus Wiseman, Mar 09 2019 LINKS Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe) Erich Friedman, Illustration of initial terms FORMULA a(n) = p(n-1) - p(n-2) - p(n-5) + p(n-7) + ... + (-1)^k*(p(n-(3*k^2-k)/2) - p(n-(3*k^2+k)/2)) + ..., where p() is A000041(). E.g., A047993 a(20) = p(19) - p(18) - p(15) + p(13) + p(8) - p(5) = 490 - 385 - 176 + 101 + 22 - 7 = 45. - Vladeta Jovovic, Aug 04 2004 G.f.: Sum_{k>=1} (-1)^k * ( x^((3*k^2+k)/2) - x^((3*k^2-k)/2) ) ) / Product_{k>=1} (1-x^k). - Vladeta Jovovic, Aug 05 2004 a(n) ~ exp(Pi*sqrt(2*n/3))*Pi / (48*sqrt(2)*n^(3/2)) ~ p(n) * Pi / (4*sqrt(6*n)), where p(n) is the partition function A000041. - Vaclav Kotesovec, Oct 06 2016 EXAMPLE a(12) = 7 because the partitions of 12 where the largest part equals the number of parts are 2 + 3 + 3 + 4, 2 + 2 + 4 + 4, 1 + 3 + 4 + 4, 1 + 2 + 2 + 2 + 5, 1 + 1 + 2 + 3 + 5, 1 + 1 + 1 + 4 + 5, and 1 + 1 + 1 + 1 + 2 + 6. [Joerg Arndt, Oct 08 2012] From Gus Wiseman, Mar 09 2019: (Start) The a(1) = 1 through a(13) = 11 integer partitions:   1  21  22  311  321  322   332   333    4222   4322    4332    4333                        331   4211  4221   4321   4331    4422    4432                        4111        4311   4411   4421    4431    4441                                    51111  52111  52211   52221   52222                                                  53111   53211   53221                                                  611111  54111   53311                                                          621111  54211                                                                  55111                                                                  622111                                                                  631111                                                                  7111111 (End) MAPLE with(combinat): for n from 1 to 36 do P:=partition(n): c:=0: for j from 1 to nops(P) do if P[j][nops(P[j])]=nops(P[j]) then c:=c+1 else c:=c fi od: a[n]:=c: od: seq(a[n], n=1..36); # Emeric Deutsch, Dec 11 2004 MATHEMATICA Table[ Count[Partitions[n], par_List/; First[par]===Length[par]], {n, 12}] or recur: Sum[T[n-(2m-1), m-1, m-1], {m, Ceiling[Sqrt[n]], Floor[(n+1)/2]}] with T[m_, a_, b_]/; b < a := T[m, b, a]; T[m_, a_, b_]/; m > a*b := 0; T[m_, a_, b_]/; (2m > a*b) := T[a*b-m, a, b]; T[m_, 1, b_] := If[b < m, 0, 1]; T[0, _, _] := 1; T[m_, a_, b_] := T[m, a, b]=Sum[T[m-a*i, a-1, b-i], {i, 0, Floor[m/a]}]; Table[Sum[ -(-1)^k*(p[n-(3*k^2-k)/2] - p[n-(3*k^2+k)/2]), {k, 1, Floor[(1+Sqrt[1+24*n])/6]}] /. p -> PartitionsP, {n, 1, 64}] (* Wouter Meeussen *) (* also *) Table[Count[IntegerPartitions[n], q_ /; Max[q] == Length[q]], {n, 24}] (* Clark Kimberling, Feb 13 2014 *) PROG (PARI) N=66;  q='q + O('q^N ); S=2+2*ceil(sqrt(N)); gf= sum(k=1, S,  (-1)^k * ( q^((3*k^2+k)/2) - q^((3*k^2-k)/2) ) ) / prod(k=1, N, 1-q^k ); /* Joerg Arndt, Oct 08 2012 */ (Haskell) a047993 = flip a063995 0  -- Reinhard Zumkeller, Jul 24 2013 CROSSREFS Cf. A000700, A063995, A064173, A064174. Cf. A003114, A006141, A039900, A090858, A106529, A324516, A324518, A324520. Sequence in context: A145815 A059851 A327637 * A033177 A175512 A240829 Adjacent sequences:  A047990 A047991 A047992 * A047994 A047995 A047996 KEYWORD nice,nonn AUTHOR STATUS approved

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Last modified April 5 16:53 EDT 2020. Contains 333245 sequences. (Running on oeis4.)