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 A195012 #64 Jul 06 2025 09:19:32
%S A195012 1,1,1,2,2,4,5,7,10,13,17,24,31,40,53,69,88,113,144,183,231,290,362,
%T A195012 453,563,696,859,1058,1296,1587,1935,2354,2856,3458,4175,5033,6051,
%U A195012 7259,8692,10390,12391,14756,17537,20808,24648,29151,34417,40581,47773,56158
%N A195012 Sum of positive cranks minus the sum of positive ranks of all partitions of n.
%C A195012 It appears this is also the column 0 of triangle A195011 without the first one (see the Andrews-Garvan-Liang paper, page 16).
%C A195012 Is this also the ospt(n) function mentioned in the Andrews-Chan-Kim paper? Is A115995(n) the first crank momment? Is A209616(n) the first rank moment? - Omar E. Pol, Apr 07 2012
%C A195012 From _Jeremy Lovejoy_, Oct 14 2022: (Start)
%C A195012 a(n) is also the number of rank 0 strongly unimodal sequences of size n. A strongly unimodal sequence is a sequence of positive integers which are strictly increasing up to a point (the peak) and then strictly decreasing thereafter. The size is the sum of all of the parts and the rank is the number of parts to the left of the peak minus the number of parts to the right of the peak.
%C A195012 For example, there are 10 strongly unimodal sequences of size 6: (6), (1,5), (5,1), (2,4), (4,2), (1,4,1), (3,2,1), (1,2,3), (1,3,2), and (2,3,1). The sequences (6), (1,4,1), (1,3,2), and (2,3,1) have rank 0, and so a(6) = 4. (End)
%H A195012 G. E. Andrews, S. H. G. Chan, and B. Kim, <a href="http://www.math.psu.edu/andrews/pdf/292.pdf">The odd moments of ranks and cranks</a> (This is the function C_1 - R_1 of that paper), Journal of Combinatorial Theory, Series A, Volume 120, Issue 1, January 2013, Pages 77-91.
%H A195012 G. E. Andrews, F. G. Garvan, and J. Liang, <a href="http://www.math.psu.edu/andrews/pdf/287.pdf">Combinatorial interpretation of congruences for the spt-function</a>, The Ramanujan Journal, December 2012, Volume 29, Issue 1-3, pp 321-338.
%H A195012 A. O. L. Atkin and F. G. Garvan, <a href="https://arxiv.org/abs/math/0208050">Relations between the ranks and cranks of partitions</a>, arXiv:math/0208050 [math.NT], 2002.
%H A195012 K. Bringmann, C. Jennings-Shaffer, K. Mahlburg, and R. Rhoades, <a href="https://doi.org/10.1090/tran/7791">Peak positions of strongly unimodal sequences</a>, Trans. Amer. Math. Soc. 372 (2019), 7087-7109.
%H A195012 Frank Garvan, <a href="http://www.combinatorics.net/conf/A75/Slides/02_03_Garvan.pdf">Dyson's rank function and Andrews's SPT-function</a> [Broken link?]
%H A195012 K. Hikami and J. Lovejoy, <a href="https://doi.org/10.1186/s40687-014-0016-3">Torus knots and quantum modular forms</a>, Res. Math. Sci. 2, Article 2 (2015).
%F A195012 a(n) = A115995(n) - A209616(n).
%F A195012 From _Jeremy Lovejoy_, Oct 14 2022: (Start)
%F A195012 G.f.: (1/Product_{n>=1}(1-x^n))*Sum_{n>=1} x^(n*(n+1)/2)*(-1)^(n-1)*(1-x^(n^2))/(1-x^n).
%F A195012 G.f.: (1/Product_{n>=1}(1-x^n))*Sum_{n,r>=0} (-1)^(n+r)*x^(n*(3*n+5)/2+2*n*r+r*(r+3)/2). (End)
%F A195012 a(n) ~ exp(Pi*sqrt(2*n/3)) / (16*sqrt(3)*n). - _Vaclav Kotesovec_, Jul 06 2025
%e A195012 For n = 6 we have:
%e A195012 ------------------------------------------------
%e A195012 Partitions
%e A195012 of 6 Crank Rank
%e A195012 ------------------------------------------------
%e A195012 6 6 6 - 1 = 5
%e A195012 3+3 3 3 - 2 = 1
%e A195012 4+2 4 4 - 2 = 2
%e A195012 2+2+2 2 2 - 3 = -1
%e A195012 5+1 1 - 1 = 0 5 - 2 = 3
%e A195012 3+2+1 2 - 1 = 1 3 - 3 = 0
%e A195012 4+1+1 1 - 2 = -1 4 - 3 = 1
%e A195012 2+2+1+1 0 - 2 = -2 2 - 4 = -2
%e A195012 3+1+1+1 0 - 3 = -3 3 - 4 = -1
%e A195012 2+1+1+1+1 0 - 4 = -4 2 - 5 = -3
%e A195012 1+1+1+1+1+1 0 - 6 = -6 1 - 6 = -5
%e A195012 ------------------------------------------------
%e A195012 The sum of positive cranks is 6+3+4+2+1 = 16 and the sum of positive ranks is 5+1+2+3+1 = 12 therefore a(6) = 16 - 12 = 4.
%p A195012 # Based on Theorem 1 of Andrews-Chan-Kim:
%p A195012 M:=101;
%p A195012 qinf:=mul(1-q^i,i=1..M);
%p A195012 qinf:=series(qinf,q,M);
%p A195012 C1:=add((-1)^(n+1)*q^(n*(n+1)/2)/(1-q^n),n=1..M);
%p A195012 C1:=series(C1/qinf,q,M);
%p A195012 R1:=add((-1)^(n+1)*q^(n*(3*n+1)/2)/(1-q^n),n=1..M);
%p A195012 R1:=series(R1/qinf,q,M);
%p A195012 series(C1-R1,q,M);
%p A195012 seriestolist(%); # _N. J. A. Sloane_, Sep 04 2012
%t A195012 M = 101;
%t A195012 qinf = Product[1-q^i, {i, 1, M}];
%t A195012 qinf = Series[qinf, {q, 0, M}];
%t A195012 C1 = Sum[(-1)^(n+1) q^(n(n+1)/2)/(1-q^n), {n, 1, M}];
%t A195012 C1 = Series[C1/qinf, {q, 0, M}];
%t A195012 R1 = Sum[(-1)^(n+1) q^(n(3n+1)/2)/(1-q^n), {n, 1, M}];
%t A195012 R1 = Series[R1/qinf, {q, 0, M}];
%t A195012 CoefficientList[Series[C1-R1, {q, 0, M}], q] // Rest (* _Jean-François Alcover_, Aug 18 2018, translated from Maple *)
%Y A195012 Cf. A092269, A115995, A195011, A208482, A209616.
%Y A195012 Cf. A059618.
%K A195012 nonn
%O A195012 1,4
%A A195012 _Omar E. Pol_, Jan 10 2012
%E A195012 New name, example and more terms from _Omar E. Pol_, Apr 06 2012
%E A195012 More terms a(44)-a(50) from _Alois P. Heinz_, Apr 08 2012