OFFSET
1,6
COMMENTS
Column k=1 in the triangle A063995.
REFERENCES
George E. Andrews, The Theory of Partitions, Addison-Wesley, Reading, Mass., 1976.
LINKS
Seiichi Manyama, Table of n, a(n) for n = 1..10000
FORMULA
G.f. for the number of partitions of n with rank r is Sum((-1)^k*x^(r*k)*(x^((3*k^2+k)/2)-x^((3*k^2-k)/2)), k=1..infinity)/Product(1-x^k, k=1..infinity). - Vladeta Jovovic, Dec 20 2004
Also Sum(x^(2*n+r+1)*Product((1-x^(2*n+r+1-k))/(1-x^k),k=1..n),n=0..infinity). - Vladeta Jovovic, May 05 2008
a(n) ~ Pi * exp(Pi*sqrt(2*n/3)) / (3 * 2^(9/2) * n^(3/2)). - Vaclav Kotesovec, May 26 2023
EXAMPLE
a(6)=2 because the 11 partitions 6,51,42,411,33,321,3111,222,2211,21111,111111 have ranks 5,3,2,1,1,0,-1,-1,-2,-3,-5, respectively.
MAPLE
with(combinat): for n from 1 to 35 do P:=partition(n): c:=0: for j from 1 to nops(P) do if P[j][nops(P[j])]-nops(P[j])=1 then c:=c+1 else c:=c fi od: a[n]:=c: od: seq(a[n], n=1..35);
MATHEMATICA
Table[Count[IntegerPartitions[n], _?(Max[#]-Length[#]==1&)], {n, 60}] (* Harvey P. Dale, Nov 29 2014 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Dec 12 2004
STATUS
approved