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A209616 Sum of positive Dyson's ranks of all partitions of n.

%I A209616 #53 Jul 06 2025 08:55:10
%S A209616 0,1,2,4,7,12,18,29,42,63,89,128,176,246,333,453,603,807,1058,1393,
%T A209616 1807,2346,3011,3867,4915,6248,7879,9926,12421,15529,19297,23954,
%U A209616 29585,36486,44802,54937,67096,81831,99459,120700,146026,176410,212512,255636,306734
%N A209616 Sum of positive Dyson's ranks of all partitions of n.
%C A209616 The Dyson's rank of a partition is the largest part minus the number of parts.
%H A209616 Seiichi Manyama, <a href="/A209616/b209616.txt">Table of n, a(n) for n = 1..10000</a>
%H A209616 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> (See the function R_1), Journal of Combinatorial Theory, Series A, Volume 120, Issue 1, January 2013, Pages 77-91.
%H A209616 F. J. Dyson, <a href="https://archim.org.uk/eureka/archive/Eureka-8.pdf">Some guesses in the theory of partitions</a>, Eureka (Cambridge) 8 (1944), 10-15.
%H A209616 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>
%F A209616 a(n) = A115995(n) - A195012(n). - _Omar E. Pol_, Apr 06 2012
%F A209616 G.f.: (1/Product_{k>=1} (1-x^k)) * Sum_{k>=1} (-1)^(k-1) * x^(k*(3*k+1)/2) / (1-x^k). - _Seiichi Manyama_, May 21 2023
%F A209616 a(n) ~ log(2) * exp(Pi*sqrt(2*n/3)) / (Pi*2^(3/2)*sqrt(n)). - _Vaclav Kotesovec_, Jul 06 2025
%e A209616 For n = 5 we have:
%e A209616 --------------------------
%e A209616 Partitions        Dyson's
%e A209616 of 5               rank
%e A209616 --------------------------
%e A209616 5               5 - 1 =  4
%e A209616 4+1             4 - 2 =  2
%e A209616 3+2             3 - 2 =  1
%e A209616 3+1+1           3 - 3 =  0
%e A209616 2+2+1           2 - 3 = -1
%e A209616 2+1+1+1         2 - 4 = -2
%e A209616 1+1+1+1+1       1 - 5 = -4
%e A209616 --------------------------
%e A209616 The sum of positive Dyson's ranks of all partitions of 5 is 4+2+1 = 7 so a(5) = 7.
%p A209616 # Maple program based on Theorem 1 of Andrews-Chan-Kim:
%p A209616 M:=101;
%p A209616 qinf:=mul(1-q^i,i=1..M);
%p A209616 qinf:=series(qinf,q,M);
%p A209616 R1:=add((-1)^(n+1)*q^(n*(3*n+1)/2)/(1-q^n),n=1..M);
%p A209616 R1:=series(R1/qinf,q,M);
%p A209616 seriestolist(%); # _N. J. A. Sloane_, Sep 04 2012
%t A209616 M = 101;
%t A209616 qinf = Product[1-q^i, {i, 1, M}];
%t A209616 qinf = Series[qinf, {q, 0, M}];
%t A209616 R1 = Sum[(-1)^(n+1) q^(n(3n+1)/2)/(1-q^n), {n, 1, M}];
%t A209616 R1 = Series[R1/qinf, {q, 0, M}];
%t A209616 CoefficientList[R1, q] // Rest (* _Jean-François Alcover_, Aug 18 2018, translated from Maple *)
%o A209616 (PARI) my(N=50, x='x+O('x^N)); concat(0, Vec(1/prod(k=1, N, 1-x^k)*sum(k=1, N, (-1)^(k-1)*x^(k*(3*k+1)/2)/(1-x^k)))) \\ _Seiichi Manyama_, May 21 2023
%Y A209616 Column 1 of triangle A208482.
%Y A209616 Cf. A063995, A064174, A092269, A105805, A194547, A194549, A195822, A208478.
%K A209616 nonn
%O A209616 1,3
%A A209616 _Omar E. Pol_, Mar 10 2012
%E A209616 More terms from _Alois P. Heinz_, Mar 10 2012