|
| |
|
|
A115995
|
|
Sum of the sizes of the Durfee squares of all partitions of n.
|
|
8
| |
|
|
0, 1, 2, 3, 6, 9, 16, 23, 36, 52, 76, 106, 152, 207, 286, 386, 522, 691, 920, 1202, 1576, 2038, 2636, 3373, 4320, 5478, 6944, 8738, 10984, 13717, 17116, 21232, 26308, 32441, 39944, 48977, 59970, 73147, 89090, 108151, 131090, 158417, 191166, 230049
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,3
|
|
|
COMMENTS
| Also sum of positive cranks of all partitions of n, n>1; see A064391. - Vladeta Jovovic (vladeta(AT)eunet.rs), Oct 20 2006
|
|
|
REFERENCES
| G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976 (pp. 27-28).
G. E. Andrews and K. Eriksson, Integer Partitions, Cambridge Univ. Press, 2004 (pp. 75-78).
G. E. Andrews and F. G. Garvan, Self-conjugate vector partitions and the parity of the spt-function, http://www.math.ufl.edu/~fgarvan/papers/spt-parity.pdf.
|
|
|
LINKS
| Eric Weisstein's World of Mathematics, Durfee Square.
|
|
|
FORMULA
| G.f.=sum(kz^(k^2)/product((1-z^j)^2,j=1..k),k=1..infinity).
a(n) = sum_{k=1}^{floor(sqrt(n))} k*A115994(n,k).
Convolution of A067742 and A000041. - Vladeta Jovovic (vladeta(AT)eunet.rs), Oct 20 2006
|
|
|
EXAMPLE
| a(4)=6 because the partitions [4],[3,1],[2,2],[2,1,1] and
[1,1,1,1] of 4 have Durfee squares of sizes 1,1,2,1 and 1, respectively.
|
|
|
MAPLE
| g:=sum(k*z^(k^2)/product((1-z^j)^2, j=1..k), k=1..10): gser:=series(g, z=0, 56): seq(coeff(gser, z, n), n=1..52);
|
|
|
CROSSREFS
| Cf. A115994, A115720, A115721, A115722.
Sequence in context: A067435 A035494 A003244 * A051057 A147364 A147227
Adjacent sequences: A115992 A115993 A115994 * A115996 A115997 A115998
|
|
|
KEYWORD
| nonn
|
|
|
AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 11 2006
|
|
|
EXTENSIONS
| Edited and verified by Frank Adams-Watters (FrankTAW(AT)Netscape net) Mar 11 2006
|
| |
|
|
