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A237755
Number of partitions of n such that 2*(greatest part) >= (number of parts).
24
1, 2, 2, 4, 6, 9, 12, 18, 24, 34, 46, 63, 83, 111, 144, 190, 245, 318, 405, 520, 657, 833, 1045, 1312, 1634, 2036, 2517, 3114, 3829, 4705, 5751, 7027, 8544, 10381, 12564, 15190, 18301, 22026, 26425, 31669, 37849, 45180, 53796, 63983, 75923, 89987, 106435
OFFSET
1,2
COMMENTS
Also, the number of partitions of n such that (greatest part) <= 2*(number of parts); hence, the number of partitions of n such that (rank + greatest part) >= 0.
LINKS
FORMULA
a(n) = A000041(n) - A237751(n).
G.f.: Sum_{n>=1} x^n * Product_{k=1..n} (1 - x^(2*n+k-1))/(1 - x^k). - Paul D. Hanna, Aug 03 2015
EXAMPLE
a(6) = 9 counts all of the 11 partitions of 6 except these: 21111, 111111.
MATHEMATICA
z = 50; Table[Count[IntegerPartitions[n], p_ /; 2 Max[p] >= Length[p]], {n, z}]
PROG
(PARI) {a(n) = my(A); A = sum(m=0, n, x^m*prod(k=1, m, (1-x^(2*m+k-1))/(1-x^k +x*O(x^n)))); polcoeff(A, n)}
for(n=1, 60, print1(a(n), ", ")) \\ Paul D. Hanna, Aug 03 2015
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Feb 13 2014
STATUS
approved