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 A334651 a(n) is the total number of down steps between the first and second up steps in all 4_1-Dyck paths of length 5*n. 1
 0, 7, 25, 155, 1195, 10282, 94591, 910480, 9054965, 92310075, 959473878, 10129715890, 108327387675, 1170975480360, 12773887368040, 140445927510832, 1554748206904325, 17314584431331025, 193849445090545875, 2180550929942519685, 24632294533221865028 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS A 4_1-Dyck path is a lattice path with steps (1, 4), (1, -1) that starts and ends at y = 0 and stays above the line y = -1. For n = 1, there is no 2nd up step, a(1) = 7 enumerates the total number of down steps between the 1st up step and the end of the path. LINKS A. Asinowski, B. Hackl, and S. Selkirk, Down step statistics in generalized Dyck paths, arXiv:2007.15562 [math.CO], 2020. FORMULA a(0) = 0 and a(n) = 4*binomial(5*n, n)/(n+1) - 3*binomial(5*n+1, n)/(n+1) + 8*binomial(5*(n-1), n-1)/n - 2*[n=1] for n > 0, where [ ] is the Iverson bracket. EXAMPLE For n = 1, the 4_1-Dyck paths are DUDDD, UDDDD. This corresponds to a(1) = 3 + 4 = 7 down steps between the 1st up step and the end of the path. MATHEMATICA a[0] = 0; a[n_] := 4 * Binomial[5*n, n]/(n + 1) - 3 * Binomial[5*n + 1, n]/(n + 1) + 8*Binomial[5*(n - 1), n - 1]/n - 2 * Boole[n == 1]; Array[a, 21, 0] (* Amiram Eldar, May 13 2020 *) PROG (SageMath) [4*binomial(5*n, n)/(n + 1) - 3*binomial(5*n + 1, n)/(n + 1) + 8*binomial(5*(n - 1), n - 1)/n - 2*(n==1) if n > 0 else 0 for n in srange(30)] CROSSREFS Cf. A002294, A124724, A334642, A334647, A334719, A334786, A334787. Sequence in context: A241714 A151491 A208425 * A191237 A088009 A293532 Adjacent sequences:  A334648 A334649 A334650 * A334652 A334653 A334654 KEYWORD nonn,easy AUTHOR Benjamin Hackl, May 13 2020 STATUS approved

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Last modified December 7 19:54 EST 2021. Contains 349585 sequences. (Running on oeis4.)