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A363006 a(n) = 1/((d-1)*n + 1)*Sum_{i=0..n} binomial((d - 1)*n+1, n-i) * binomial((d-1)*n+i, i), with d = 6. 6
1, 2, 22, 342, 6202, 122762, 2571326, 56031470, 1257199154, 28849835538, 673953255142, 15973925161030, 383186776643946, 9285457458463770, 226959074854361742, 5588974707042304222, 138529985051020001634, 3453373395317346136610, 86526667346028323084726, 2177844556015530807952438 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
See Yang-Jiang paper, related to large Schröder numbers, which correspond to the formula in the Name, instead with d=2.
LINKS
Lin Yang, Yu-Yuan Zhang, and Sheng-Liang Yang, The halves of Delannoy matrix and Chung-Feller properties of the m-Schröder paths, Linear Alg. Appl. (2024).
Sheng-liang Yang and Mei-yang Jiang, Pattern avoiding problems on the hybrid d-trees, J. Lanzhou Univ. Tech., (China, 2023) Vol. 49, No. 2, 144-150. See p. 145. (in Mandarin.)
FORMULA
G.f. satisfies A(x) = 1 + x * A(x)^5 * (1 + A(x)). - Seiichi Manyama, May 29 2023
From Seiichi Manyama, Aug 09 2023: (Start)
a(n) = (1/n) * Sum_{k=0..n-1} (-1)^k * 2^(n-k) * binomial(n,k) * binomial(6*n-k,n-1-k) for n > 0.
a(n) = (1/n) * Sum_{k=1..n} 2^k * binomial(n,k) * binomial(5*n,k-1) for n > 0. (End)
MATHEMATICA
With[{d = 6}, Table[(1/((d - 1) n + 1)) Sum[Binomial[(d - 1) n + 1, n - i] Binomial[(d - 1) n + i, i], {i, 0, n}], {n, 0, 12}] ]
PROG
(PARI) a(n) = my(d=6); sum(i=0, n, binomial((d - 1)*n+1, n-i) * binomial((d-1)*n+i, i))/((d-1)*n + 1); \\ Michel Marcus, May 16 2023
CROSSREFS
Cf. A006318 (d=2), A027307 (d=3), A144097 (d=4), A260332 (d=5).
Sequence in context: A279619 A364826 A245113 * A363304 A078232 A151615
KEYWORD
nonn
AUTHOR
Michael De Vlieger, May 16 2023
STATUS
approved

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Last modified February 20 21:52 EST 2024. Contains 370219 sequences. (Running on oeis4.)