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 A006141 M0260 #136 Feb 16 2025 08:32:29
%S A006141 1,0,0,1,1,1,1,1,2,2,3,3,4,4,5,6,7,8,10,11,13,15,17,19,23,25,29,33,38,
%T A006141 42,49,54,62,69,78,87,99,109,123,137,154,170,191,211,236,261,290,320,
%U A006141 357,392,435,479,530,582,644,706,779,854,940,1029,1133,1237,1358,1485
%N A006141 Number of integer partitions of n whose smallest part is equal to the number of parts.
%C A006141 Or, number of partitions of n in which number of largest parts is equal to the largest part.
%C A006141 a(n) is the number of partitions of n-1 without parts that differ by less than 2 and which have no parts less than three. [MacMahon]
%C A006141 There are two conflicting choices for the offset in this sequence. For the definition given here the offset is 1, and that is what we shall adopt. On the other hand, if one arrives at this sequence via the Rogers-Ramanujan identities (see the next comment), the natural offset is 0.
%C A006141 Related to Rogers-Ramanujan identities: Let G[1](q) and G[2](q) be the generating functions for the two Rogers-Ramanujan identities of A003114 and A003106, starting with the constant term 1. The g.f. for the present sequence is G[3](q) = (G[1](q) - G[2](q))/q = 1+q^3+q^4+q^5+q^6+q^7+2*q^8+2*q^9+3*q^10+.... - _Joerg Arndt_, Oct 08 2012; _N. J. A. Sloane_, Nov 18 2015
%C A006141 For more about the generalized Rogers-Ramanujan series G[i](x) see the Andrews-Baxter and Lepowsky-Zhu papers. The present series is G[3](x). - _N. J. A. Sloane_, Nov 22 2015
%C A006141 From _Wolfdieter Lang_, Oct 31 2016: (Start)
%C A006141 From Hardy (H) p. 94, eq. (6.12.1) and Hardy-Wright (H-W), p. 293, eq. (19.14.3) for H_2(a,x) - H_1(a,x) = a*H_1(a*x,x) one finds from the result for H_1(a,x) (in (H) on top on p. 95), after putting a=x, the o.g.f. of a(n) = A003114(n) - A003106(n), n >= 0, with a(0) = 0 as Sum_{m>=0} x^((m+1)^2) / Product_{j=1..m} (1 - x^j). The m=0 term is 1*x^1. See the formula given by _Joerg Arndt_, Jan 29 2011.
%C A006141 This formula has a combinatorial interpretation (found similar to the one given in (H) section 6.0, pp. 91-92 or (H-W) pp. 290-291): a(n) is the number of partitions of n with parts differing by at least 2 and part 1 present. See the example for a(15) below. (End)
%C A006141 The Heinz numbers of these integer partitions are given by A324522. - _Gus Wiseman_, Mar 09 2019
%D A006141 G. H. Hardy, Ramanujan, AMS Chelsea Publ., Providence, RI, 2002, pp. 92-95.
%D A006141 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth ed., Clarendon Press, Oxford, 2003, pp. 292-294.
%D A006141 P. A. MacMahon, Combinatory Analysis, Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 45, Section 293.
%D A006141 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006141 Alois P. Heinz, <a href="/A006141/b006141.txt">Table of n, a(n) for n = 1..1000</a>
%H A006141 George E. Andrews and R. J. Baxter, <a href="http://www.computing-wisdom.com/jstor/rogers-ramanujan.pdf">A motivated proof of the Rogers-Ramanujan identities</a>, Amer. Math. Monthly 96 (1989), no. 5, 401-409.
%H A006141 Shashank Kanade, <a href="http://www.math.rutgers.edu/~skanade/SK-Defense-Handout.pdf">Some results on the representation theory of vertex operator algebras and integer partition identities</a>, PhD Handout, Math. Dept., Rutgers University, April 2015.
%H A006141 Shashank Kanade, <a href="http://dx.doi.org/doi:10.7282/T3TX3H7B">Some results on the representation theory of vertex operator algebras and integer partition identities</a>, PhD Dissertation, Math. Dept., Rutgers University, April 2015.
%H A006141 James Lepowsky and Minxian Zhu, <a href="http://arxiv.org/abs/1205.6570">A motivated proof of Gordon's identities</a>, arXiv:1205.6570 [math.CO], 2012; The Ramanujan Journal 29.1-3 (2012): 199-211.
%H A006141 Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H A006141 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H A006141 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Rogers-RamanujanIdentities.html">Rogers-Ramanujan Identities</a>
%F A006141 G.f.: Sum_{m>=1} (x^(m^2)-x^(m*(m+1))) / Product_{i=1..m} (1-x^i) .
%F A006141 G.f.: Sum_{n>=1} x^(n^2)/Product_{k=1..n-1} (1-x^k). - _Joerg Arndt_, Jan 29 2011
%F A006141 a(n) = A003114(n) - A003106(n) = A039900(n) - A039899(n), (offset 1). - _Vladeta Jovovic_, Jul 17 2004
%F A006141 Plouffe in his 1992 dissertation conjectured that this has g.f. = (1+z+z^4+2*z^5-z^3-z^8+3*z^10-z^7+z^9)/(1+z-z^4-2*z^3-z^8+z^10), but _Michael Somos_ pointed out on Jan 22 2008 that this is false.
%F A006141 Expansion of ( f(-x^2, -x^3) - f(-x, -x^4) ) / f(-x) in powers of x where f(, ) is Ramanujan's general theta function. - _Michael Somos_, Jan 22 2007
%F A006141 a(n) ~ sqrt(1/sqrt(5) - 2/5) * exp(2*Pi*sqrt(n/15)) / (2*3^(1/4)*n^(3/4)). - _Vaclav Kotesovec_, Nov 01 2016
%e A006141 G.f. = x + x^4 + x^5 + x^6 + x^7 + x^8 + 2*x^9 + 2*x^10 + 3*x^11 + 3*x^12 + ...
%e A006141 a(15) = 5 because the partitions of 15 where the smallest part equals the number of parts are
%e A006141 3 + 6 + 6,
%e A006141 3 + 5 + 7,
%e A006141 3 + 4 + 8,
%e A006141 3 + 3 + 9, and
%e A006141 2 + 13.
%e A006141 - _Joerg Arndt_, Oct 08 2012
%e A006141 a(15) = 5 because the partitions of 15 with parts differing by at least 2 and part 1 present are: [14,1] obtained from the partition of 11 with one part, [11], added to the first part of the special partition [3,1] of 4 and [11,3,1], [10,4,1], [9,5,1], [8,6,1] from adding all partition of 15 - 9 = 6 with one part, [6], and those with two parts, [5,1], [4,1], [3,3], to the special partition [5,3,1] of 9. - _Wolfdieter Lang_, Oct 31 2016
%e A006141 a(15) = 5 because the partitions of 14 with parts >= 3 and parts differing by at least 2 are [14], [11,3], [10,4], [9,5] and [8,6]. See the second [MacMahon] comment. This follows from the g.f. G[3](q) given in Andrews - Baxter, eq. (5.1) for i=3, (using summation index m) and m*(m+2) = 3 + 5 + ... + (2*m+1). - _Wolfdieter Lang_, Nov 02 2016
%e A006141 From _Gus Wiseman_, Mar 09 2019: (Start)
%e A006141 The a(8) = 1 through a(15) = 5 integer partitions:
%e A006141 (6,2) (7,2) (8,2) (9,2) (10,2) (11,2) (12,2) (13,2)
%e A006141 (3,3,3) (4,3,3) (4,4,3) (5,4,3) (5,5,3) (6,5,3) (6,6,3)
%e A006141 (5,3,3) (6,3,3) (6,4,3) (7,4,3) (7,5,3)
%e A006141 (7,3,3) (8,3,3) (8,4,3)
%e A006141 (9,3,3)
%e A006141 (End)
%p A006141 b:= proc(n, i) option remember; `if`(n<0, 0, `if`(n=0, 1,
%p A006141 `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i,i)))))
%p A006141 end:
%p A006141 a:= n-> add(b(n-j^2, j-1), j=0..isqrt(n)):
%p A006141 seq(a(n), n=1..80); # _Alois P. Heinz_, Oct 08 2012
%t A006141 b[n_, i_] := b[n, i] = If[n<0, 0, If[n == 0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, b[n-i, i]]]]]; a[n_] := Sum[b[n-j^2, j-1], {j, 0, Sqrt[n]}]; Table[a[n], {n, 1, 80}] (* _Jean-François Alcover_, Mar 17 2014, after _Alois P. Heinz_ *)
%t A006141 Table[Length[Select[IntegerPartitions[n],Min[#]==Length[#]&]],{n,30}] (* _Gus Wiseman_, Mar 09 2019 *)
%o A006141 (PARI) {a(n) = if( n<1, 0, polcoeff( sum(k=1, sqrtint(n), x^k^2 / prod(j=1, k-1, 1 - x^j, 1 + O(x ^ (n - k^2 + 1) ))), n))} /* _Michael Somos_, Jan 22 2008 */
%Y A006141 For the generalized Rogers-Ramanujan series G[1], G[2], G[3], G[4], G[5], G[6], G[7], G[8] see A003114, A003106, A006141, A264591, A264592, A264593, A264594, A264595. G[0] = G[1]+G[2] is given by A003113.
%Y A006141 Cf. A064174, A090858, A324516, A324518, A324520, A324522.
%Y A006141 A003106 counts partitions with minimum > length.
%Y A006141 A003114 counts partitions with minimum >= length.
%Y A006141 A026794 counts partitions by minimum.
%Y A006141 A039899 counts partitions with minimum < length.
%Y A006141 A039900 counts partitions with minimum <= length.
%Y A006141 A239950 counts partitions with minimum equal to number of distinct parts.
%Y A006141 Sequences related to balance:
%Y A006141 - A010054 counts balanced strict partitions.
%Y A006141 - A047993 counts balanced partitions.
%Y A006141 - A098124 counts balanced compositions.
%Y A006141 - A106529 ranks balanced partitions.
%Y A006141 - A340596 counts co-balanced factorizations.
%Y A006141 - A340598 counts balanced set partitions.
%Y A006141 - A340599 counts alt-balanced factorizations.
%Y A006141 - A340600 counts unlabeled balanced multiset partitions.
%Y A006141 - A340653 counts balanced factorizations.
%Y A006141 Cf. A055396, A114638, A117409, A325134, A340601, A340611, A340654, A340655.
%K A006141 nonn
%O A006141 1,9
%A A006141 _N. J. A. Sloane_
%E A006141 More terms from Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jun 20 2000
%E A006141 Better description from _Naohiro Nomoto_, Feb 06 2002
%E A006141 Name shortened by _Gus Wiseman_, Apr 07 2021 (balanced partitions are A047993).