A344394 a(n) = binomial(n, n/2 - 1/4 + (-1)^n/4)*hypergeom([-n/4 - 1/8 + (-1)^n/8, -n/4 + 3/8 + (-1)^n/8], [n/2 + 7/4 + (-1)^n/4], 4).

1, 1, 2, 5, 9, 25, 44, 133, 230, 726, 1242, 4037, 6853, 22737, 38376, 129285, 217242, 740554, 1239980, 4266830, 7123765, 24701425, 41141916, 143567173, 238637282, 837212650, 1389206210, 4896136845, 8112107475, 28703894775, 47495492400, 168640510725, 278722764954

Offset: 0

Author

Peter Luschny, May 19 2021

Keywords

• nonn

Comments

Related to the Motzkin triangle A064189 counting certain lattice paths.

Crossrefs

Cf. A026300, A064189, A026302 (even bisection), A344396 (odd bisection), A327871.

Programs

Maple

alias(C=binomial):
a := n -> add(C(n, j)*(C(n - j, j + n/2 - 1/4 + (-1)^n/4) - C(n - j, j + n/2 + 7/4 + (-1)^n/4)), j = 0..n): seq(a(n), n = 0..32);

Mathematica

a[n_] := Binomial[n, n/2 - 1/4 + (-1)^n/4] Hypergeometric2F1[-n/4 - 1/8 + (-1)^n/8, -n/4 + 3/8 + (-1)^n/8, n/2 + 7/4 + (-1)^n/4, 4];
Table[a[n], {n, 0, 32}]

Formula

a(n) = Sum_{j = 0..n} C(n, j)*(C(n - j, j + n/2 - 1/4 + (-1)^n/4) - C(n - j, j + n/2 + 7/4 + (-1)^n/4)).

a(n) = A064189(n, floor(n/2)), the middle column of the Motzkin triangle.

a(n) = A026300(n, ceiling(n/2)).