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A209616
Sum of positive Dyson's ranks of all partitions of n.
17
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
1,3
COMMENTS
The Dyson's rank of a partition is the largest part minus the number of parts.
LINKS
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.
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
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
Column 1 of triangle A208482.
Sequence in context: A035296 A230118 A105807 * A192521 A066699 A188425
KEYWORD
nonn
AUTHOR
Omar E. Pol, Mar 10 2012
EXTENSIONS
More terms from Alois P. Heinz, Mar 10 2012
STATUS
approved