A089466 Inverse hyperbinomial transform of A089467.

1, 1, 3, 18, 163, 1950, 28821, 505876, 10270569, 236644092, 6098971555, 173823708696, 5427760272507, 184267682837992, 6757353631762293, 266191329601854000, 11210291102456374801, 502602430218071545104, 23900770928782913595651, 1201581698963550283673632

Offset

0,3

Comments

See A088956 for the definition of the hyperbinomial transform.

a(n) is the number of functions f:{1,2,...,n}->{1,2,...,n} such that the functional digraph contains no cycles of length 2. - Geoffrey Critzer, Mar 21 2012

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Formula

A089467(n) = Sum_{k=0..n} (n-k+1)^(n-k-1)*C(n, k)*a(k).

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

E.g.f.: exp(-(T(x))^2/2)/(1-T(x)), where T(x) is the e.g.f. for A000169. - Geoffrey Critzer, Mar 21 2012

a(n) ~ exp(-1/2) * n^n. - Vaclav Kotesovec, Oct 08 2013

a(n) = n! * Sum_{k=0..floor(n/2)} (-1/2)^k * n^(n - 2*k) / (k! * (n - 2*k)!). - Daniel Suteu, Jun 19 2018

Mathematica

nn=20; t=Sum[n^(n-1)x^n/n!, {n, 1, nn}]; a=Log[1/(1-t)]; Range[0, nn]! CoefficientList[Series[Exp[a-t^2/2], {x, 0, nn}], x] (* Geoffrey Critzer, Mar 21 2012 *)

Prog

(PARI) a(n)=if(n<0, 0, sum(m=0, n, sum(j=0, m, binomial(m, j)*binomial(n, n-m-j)*(n-1)^(n-m-j)*(m+j)!/(-2)^j)/m!))
(PARI) a(n) = n! * sum(k=0, n\2, (-1/2)^k * n^(n - 2*k) / (k! * (n - 2*k)!)); \\ Daniel Suteu, Jun 19 2018

Crossrefs

Cf. A089467, A088956.

Sequence in context: A065058 A032031 A127646 * A302585 A107403 A319938

Adjacent sequences: A089463 A089464 A089465 * A089467 A089468 A089469

Keyword

nonn,changed

Author

Paul D. Hanna, Nov 08 2003

Status

approved