This site is supported by donations to The OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A209616 Sum of positive Dyson's ranks of all partitions of n. 14
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS The Dyson's rank of a partition is the largest part minus the number of parts. REFERENCES F. J. Dyson, Some guesses in the theory of partitions, Eureka (Cambridge) 8 (1944), 10-15. 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. Frank Garvan, Dyson's rank function and Andrews's SPT-function FORMULA a(n) = A115995(n) - A195012(n). - Omar E. Pol, Apr 06 2012 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 *) CROSSREFS Column 1 of triangle A208482. Cf. A063995, 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified October 20 02:54 EDT 2019. Contains 328244 sequences. (Running on oeis4.)