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 A249914 Number of partitions of 4n with equal sums of odd and even parts. 4
 1, 1, 4, 12, 30, 70, 165, 330, 704, 1380, 2688, 4984, 9394, 16665, 29970, 52096, 90090, 152064, 257180, 423360, 697851, 1129392, 1819632, 2891520, 4583250, 7162364, 11161752, 17211180, 26427544, 40208520, 60971520, 91641748, 137290956, 204198876, 302530560 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS Alois P. Heinz, Table of n, a(n) for n = 0..1000 FORMULA a(n) = A000041(n) * A035294(n) = A000041(n) * A000009(2n). a(n) ~ exp(2*Pi*sqrt(2*n/3)) / (16*6^(3/4)*n^(7/4)). - Vaclav Kotesovec, Dec 11 2020 EXAMPLE a(0) = 1: [], the empty partition. a(1) = 1: [2,1,1]. a(2) = 4: [4,3,1], [4,1,1,1,1], [3,2,2,1], [2,2,1,1,1,1]. a(3) = 12: [6,5,1], [6,3,3], [6,3,1,1,1], [6,1,1,1,1,1,1], [5,4,2,1], [5,2,2,2,1], [4,3,3,2], [4,3,2,1,1,1], [4,2,1,1,1,1,1,1], [3,3,2,2,2], [3,2,2,2,1,1,1], [2,2,2,1,1,1,1,1,1]. MAPLE b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, b(n, i-2)+`if`(i>n, 0, b(n-i, i)))) end: a:= n-> combinat[numbpart](n) *b(2*n, 2*n-1): seq(a(n), n=0..50); MATHEMATICA b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-2]+If[i>n, 0, b[n-i, i]]]]; a[n_] := PartitionsP[n] b[2n, 2n-1]; a /@ Range[0, 50] (* Jean-François Alcover, Dec 11 2020, after Alois P. Heinz *) CROSSREFS Cf. A000009, A000041, A035294, A045931, A255001. Sequence in context: A100691 A000298 A218009 * A006802 A068055 A221855 Adjacent sequences: A249911 A249912 A249913 * A249915 A249916 A249917 KEYWORD nonn AUTHOR Alois P. Heinz, Feb 11 2015 STATUS approved

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Last modified August 12 19:26 EDT 2024. Contains 375113 sequences. (Running on oeis4.)