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

1,3

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

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.

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

Sequence in context: A035296 A230118 A105807 * A192521 A066699 A188425

Adjacent sequences: A209613 A209614 A209615 * A209617 A209618 A209619

Keyword

nonn

Author

Omar E. Pol, Mar 10 2012

Extensions

More terms from Alois P. Heinz, Mar 10 2012

Status

approved