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 A047993 #117 Jan 22 2025 14:24:52
%S A047993 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,
%T A047993 196,232,285,336,414,485,591,696,839,987,1187,1389,1661,1946,2311,
%U A047993 2702,3201,3731,4400,5126,6018,6997,8195,9502,11093,12849,14949,17281,20062
%N A047993 Number of balanced partitions of n: the largest part equals the number of parts.
%C A047993 Useful in the creation of plane partitions with C3 or C3v symmetry.
%C A047993 The function T[m,a,b] used here gives the partitions of m whose Ferrers plot fits within an a X b box.
%C A047993 Central terms of triangle in A063995: a(n) = A063995(n,0). - _Reinhard Zumkeller_, Jul 24 2013
%C A047993 Sequence enumerates the collection of partitions of size n that are in the monoid 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 monoid that partitions an odd prime p or a composite number of form 2p cannot be a product of smaller nontrivial partitions, whether in this monoid or not. - _Richard Locke Peterson_, Jul 15 2018
%C A047993 The "multiplication" given above was noted earlier by _Franklin T. Adams-Watters_ in A122697. - _Richard Peterson_, Jul 19 2023
%C A047993 The Heinz numbers of these integer partitions are given by A106529. - _Gus Wiseman_, Mar 09 2019
%H A047993 Vaclav Kotesovec, <a href="/A047993/b047993.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from T. D. Noe)
%H A047993 Erich Friedman, <a href="/A047993/a047993.gif">Illustration of initial terms</a>
%F A047993 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., 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
%F A047993 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
%F A047993 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
%F A047993 G.f.: Sum_{n>=0} [2n,n]_q q^(2*n), where [m,n]_q are the q-binomial coefficients. - _Mamuka Jibladze_, Aug 12 2021
%F A047993 G.f.: Sum_{k>=1} x^(2*k-1) * Product_{j=1..k-1} (1-x^(k+j-1))/(1-x^j). - _Seiichi Manyama_, Jan 24 2022
%e A047993 From _Joerg Arndt_, Oct 08 2012: (Start)
%e A047993 a(12) = 7 because the partitions of 12 where the largest part equals the number of parts are
%e A047993 2 + 3 + 3 + 4,
%e A047993 2 + 2 + 4 + 4,
%e A047993 1 + 3 + 4 + 4,
%e A047993 1 + 2 + 2 + 2 + 5,
%e A047993 1 + 1 + 2 + 3 + 5,
%e A047993 1 + 1 + 1 + 4 + 5, and
%e A047993 1 + 1 + 1 + 1 + 2 + 6.
%e A047993 (End)
%e A047993 From _Gus Wiseman_, Mar 09 2019: (Start)
%e A047993 The a(1) = 1 through a(13) = 11 integer partitions:
%e A047993 1 21 22 311 321 322 332 333 4222 4322 4332 4333
%e A047993 331 4211 4221 4321 4331 4422 4432
%e A047993 4111 4311 4411 4421 4431 4441
%e A047993 51111 52111 52211 52221 52222
%e A047993 53111 53211 53221
%e A047993 611111 54111 53311
%e A047993 621111 54211
%e A047993 55111
%e A047993 622111
%e A047993 631111
%e A047993 7111111
%e A047993 (End)
%p A047993 A047993 := proc(n)
%p A047993 a := 0 ;
%p A047993 for p in combinat[partition](n) do
%p A047993 r := max(op(p))-nops(p) ;
%p A047993 if r = 0 then
%p A047993 a := a+1 ;
%p A047993 end if;
%p A047993 end do:
%p A047993 a ;
%p A047993 end proc:
%p A047993 seq(A047993(n),n=1..20) ; # _Emeric Deutsch_, Dec 11 2004
%t A047993 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]}];
%t A047993 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_ *)
%t A047993 (* also *)
%t A047993 Table[Count[IntegerPartitions[n], q_ /; Max[q] == Length[q]], {n, 24}]
%t A047993 (* _Clark Kimberling_, Feb 13 2014 *)
%t A047993 nmax = 100; p = 1; s = 1; Do[p = Normal[Series[p*x^2*(1 - x^(2*k - 1))*(1 + x^k)/(1 - x^k), {x, 0, nmax}]]; s += p;, {k, 1, nmax + 1}]; Take[CoefficientList[s, x], nmax] (* _Vaclav Kotesovec_, Oct 16 2024 *)
%o A047993 (PARI)
%o A047993 N=66; q='q + O('q^N );
%o A047993 S=2+2*ceil(sqrt(N));
%o A047993 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 );
%o A047993 /* _Joerg Arndt_, Oct 08 2012 */
%o A047993 (PARI) my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, x^(2*k-1)*prod(j=1, k-1, (1-x^(k+j-1))/(1-x^j)))) \\ _Seiichi Manyama_, Jan 24 2022
%o A047993 (Haskell)
%o A047993 a047993 = flip a063995 0 -- _Reinhard Zumkeller_, Jul 24 2013
%Y A047993 Cf. A000700, A063995, A064173, A064174.
%Y A047993 Cf. A003114, A006141, A039900, A090858, A106529, A324516, A324518, A324520.
%Y A047993 Column 1 of A350879.
%K A047993 nice,nonn
%O A047993 1,7
%A A047993 _Wouter Meeussen_