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 A237755 Number of partitions of n such that 2*(greatest part) >= (number of parts). 3
 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 (list; graph; refs; listen; history; text; internal format)
 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.  A237751(n) + a(n) = A000041(n). LINKS FORMULA 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 CROSSREFS Cf. A064173, A237751-A237755, A237756, A237757, A000041. Sequence in context: A319381 A035564 A240065 * A306730 A209603 A192684 Adjacent sequences:  A237752 A237753 A237754 * A237756 A237757 A237758 KEYWORD nonn,easy AUTHOR Clark Kimberling, Feb 13 2014 STATUS approved

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Last modified September 29 02:49 EDT 2020. Contains 337420 sequences. (Running on oeis4.)