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 A122880 Catalan numbers minus odd-indexed Fibonacci numbers. 11
 0, 0, 0, 1, 8, 43, 196, 820, 3265, 12615, 47840, 179355, 667875, 2478022, 9180616, 34011401, 126120212, 468411235, 1743105373, 6500874434, 24300686879, 91049069203, 341924710480, 1286932932251, 4854167659403, 18346988061078 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS From Emeric Deutsch, Aug 21 2008: (Start) Number of Dyck paths of height at least 4 and of semilength n. Example: a(5)=8 because we have UUUUUDDDDD, UUUUDUDDDD, UUUDUUDDDD, UUDUUUDDDD, UDUUUUDDDD and the reflection of the last three in a vertical axis. Number of ordered trees of height at least 4 and having n edges. (End) From Gus Wiseman, Jun 22 2019: (Start) Also the number of non-crossing, capturing set partitions of {1..n}. A set partition is crossing if it has two blocks of the form {...x...y...}, {...z...t...} where x < z < y < t or z < x < t < y, and capturing if it has two blocks of the form {...x...y...}, {...z...t...} where x < z and y > t or x > z and y < t. Capturing is a weaker condition than nesting, so for example {{1,3,5},{2,4}} is capturing but not nesting. The a(4) = 1 and a(5) = 8 non-crossing, capturing set partitions are:   {{1,4},{2,3}}  {{1,2,5},{3,4}}                  {{1,4,5},{2,3}}                  {{1,5},{2,3,4}}                  {{1},{2,5},{3,4}}                  {{1,4},{2,3},{5}}                  {{1,5},{2},{3,4}}                  {{1,5},{2,3},{4}}                  {{1,5},{2,4},{3}} (End) LINKS E. Deutsch and H. Prodinger, A bijection between directed column-convex polyominoes and ordered trees of height at most three, Theoretical Comp. Science, 307, 2003, 319-325. [Emeric Deutsch, Aug 21 2008] FORMULA A000108(n) - A001519(n), n > 0; A000108 = Catalan numbers, A001519 = odd-indexed Fibonacci numbers. EXAMPLE a(5) = 8 = A000108(5) - A001519(5) = 42 - 34. MAPLE with(combinat): seq(binomial(2*n, n)/(n+1)-fibonacci(2*n-1), n=1..27); # Emeric Deutsch, Aug 21 2008 MATHEMATICA With[{nn=30}, #[]-#[]&/@Thread[{CatalanNumber[Range[nn]], Fibonacci[ Range[ 1, 2nn, 2]]}]] (* Harvey P. Dale, Nov 07 2016 *) CROSSREFS Cf. A000108, A001519, A122881. Non-crossing set partitions are A000108. Capturing set partitions are A326243. Crossing, not capturing set partitions are A326245. Crossing, capturing set partitions are A326246. Cf. A000110, A054391, A099947, A326255, A326259. Sequence in context: A137748 A005024 A094865 * A171479 A227209 A282523 Adjacent sequences:  A122877 A122878 A122879 * A122881 A122882 A122883 KEYWORD nonn AUTHOR Gary W. Adamson, Sep 16 2006 EXTENSIONS More terms from Emeric Deutsch, Aug 21 2008 STATUS approved

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Last modified September 30 22:26 EDT 2020. Contains 337440 sequences. (Running on oeis4.)