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

0, 1, 2, 4, 7, 12, 18, 29, 42, 63, 89, 128, 176, 246, 333, 453, 603, 807, 1058, 1393, 1807, 2346, 3011, 3867, 4915, 6248, 7879, 9926, 12421, 15529, 19297, 23954, 29585, 36486, 44802, 54937, 67096, 81831, 99459, 120700, 146026, 176410, 212512, 255636, 306734

Offset: 1

Author

Omar E. Pol, Mar 10 2012

Keywords

• nonn

Comments

The Dyson's rank of a partition is the largest part minus the number of parts.

Examples

			For n = 5 we have:
--------------------------
Partitions        Dyson's
of 5               rank
--------------------------
5               5 - 1 =  4
4+1             4 - 2 =  2
3+2             3 - 2 =  1
3+1+1           3 - 3 =  0
2+2+1           2 - 3 = -1
2+1+1+1         2 - 4 = -2
1+1+1+1+1       1 - 5 = -4
--------------------------
The sum of positive Dyson's ranks of all partitions of 5 is 4+2+1 = 7 so a(5) = 7.
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000
• G. E. Andrews, S. H. G. Chan, and B. Kim, The odd moments of ranks and cranks (See the function R_1), Journal of Combinatorial Theory, Series A, Volume 120, Issue 1, January 2013, Pages 77-91.
• F. J. Dyson, Some guesses in the theory of partitions, Eureka (Cambridge) 8 (1944), 10-15.
• Frank Garvan, Dyson's rank function and Andrews's SPT-function

Crossrefs

Column 1 of triangle A208482.

Cf. A063995, A064174, A092269, A105805, A194547, A194549, A195822, A208478.

Programs

Maple

# Maple program based on Theorem 1 of Andrews-Chan-Kim:
M:=101;
qinf:=mul(1-q^i,i=1..M);
qinf:=series(qinf,q,M);
R1:=add((-1)^(n+1)*q^(n*(3*n+1)/2)/(1-q^n),n=1..M);
R1:=series(R1/qinf,q,M);
seriestolist(%); # N. J. A. Sloane, Sep 04 2012

Mathematica

M = 101;
qinf = Product[1-q^i, {i, 1, M}];
qinf = Series[qinf, {q, 0, M}];
R1 = Sum[(-1)^(n+1) q^(n(3n+1)/2)/(1-q^n), {n, 1, M}];
R1 = Series[R1/qinf, {q, 0, M}];
CoefficientList[R1, q] // Rest (* Jean-François Alcover, Aug 18 2018, translated from Maple *)

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

Formula

a(n) = A115995(n) - A195012(n). - Omar E. Pol, Apr 06 2012

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

a(n) ~ log(2) * exp(Pi*sqrt(2*n/3)) / (Pi*2^(3/2)*sqrt(n)). - Vaclav Kotesovec, Jul 06 2025

Extensions

More terms from Alois P. Heinz, Mar 10 2012