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.

1, 2, 22, 342, 6202, 122762, 2571326, 56031470, 1257199154, 28849835538, 673953255142, 15973925161030, 383186776643946, 9285457458463770, 226959074854361742, 5588974707042304222, 138529985051020001634, 3453373395317346136610, 86526667346028323084726, 2177844556015530807952438

Offset: 0

Author

Michael De Vlieger, May 16 2023

Keywords

• nonn

Comments

See Yang-Jiang paper, related to large Schröder numbers, which correspond to the formula in the Name, instead with d=2.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..500
• 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.)

Crossrefs

Cf. A006318 (d=2), A027307 (d=3), A144097 (d=4), A260332 (d=5).

Cf. A217364, A363305, A364195, A364196.

Programs

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}] ]

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

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)