OFFSET
1,3
COMMENTS
The Dyson's rank of a partition is the largest part minus the number of parts.
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
FORMULA
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
EXAMPLE
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.
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Omar E. Pol, Mar 10 2012
EXTENSIONS
More terms from Alois P. Heinz, Mar 10 2012
STATUS
approved