A350267 a(n) = n*hypergeom([1, 1 - n, -n], [2], 1).

0, 1, 4, 18, 100, 675, 5376, 49294, 510728, 5894109, 74915740, 1039180186, 15613569324, 252501251743, 4371586879128, 80652138666870, 1579212732426256, 32701859350855769, 713914404925713588, 16384896394304282722, 394340620941231415540, 9929838681717090607611

Offset

0,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..400

Lara Pudwell, Pattern Avoidance in Circular Parking Functions, Valparaiso Univ. (2026). See p. 9 (Theorem 10).

Formula

a(n) = n*A247499(n - 1) for n >= 1.

a(n) = Sum_{k=1..n} binomial(n, k)^2 * k! / (n - k + 1).

E.g.f.: (exp(x/(1-x)) - exp(x))/x. - Vladimir Kruchinin, Seiichi Manyama, Jul 01 2025

Maple

A350267 := n -> n*hypergeom([1, 1 - n, -n], [2], 1): seq(simplify(A350267(n)), n = 0..21);
# Alternative:
egf := (exp(x/(1-x)) - exp(x))/x: ser := series(egf, x, 23):
seq(n!*coeff(ser, x, n), n = 0..21);  # Peter Luschny, Jul 01 2025

Mathematica

a[n_] := Sum[Binomial[n, k]^2 * k!/(n - k + 1), {k, 1, n}]; Array[a, 22, 0] (* Amiram Eldar, Jan 09 2022 *)

Prog

(PARI) a(n) = sum(k=1, n, binomial(n, k)^2 * k! / (n - k + 1)); \\ Michel Marcus, Jan 09 2022
(Magma) [0] cat [&+[Binomial(n, k)^2 * Factorial(k)/(n-k+1): k in [1..n]]: n in [1..25]]; // Vincenzo Librandi, Sep 13 2025

Crossrefs

Row sums of A350266.

Cf. A247499.

Sequence in context: A215522 A201826 A327833 * A064852 A229286 A191365

Adjacent sequences: A350264 A350265 A350266 * A350268 A350269 A350270

Keyword

nonn

Author

Peter Luschny, Jan 09 2022

Extensions

Definition changed to a(0) = 0 by Peter Luschny, Jul 01 2025

Status

approved