A122725 a(n) = A000670(n)^2.

1, 1, 9, 169, 5625, 292681, 21930489, 2236627849, 297935847225, 50229268482121, 10454564139438969, 2632936466960600329, 789136169944454084025, 277579719258755165321161, 113238180214596650771616249, 53030348046942317338336489609, 28256184698070300360908567636025

Offset: 0

Author

Vladeta Jovovic, Sep 23 2006

Keywords

• easy
• nonn

Comments

This is also the number of possible positions of n intervals on a line having a common non-punctual intersection. Proof: Let us denoted each interval Ai (1 <= i <= n) by the string AiAi. Then the set of all such relative positions is given by the S-language [A1 ⊗ A2 ... ⊗ An]^2. The cardinality of $A1 ⊗ A2 ... ⊗ An$ is given by A000670. - Sylviane R. Schwer (schwer(AT)lipn.univ-paris13.fr), Nov 26 2007

Crossrefs

Cf. A000670, A055203, A101370.

Programs

Maple

b:= proc(n, k) option remember;
     `if`(n=0, 1, add(b(n-1, j)*j, j=k..k+1))
    end:
a:= n-> b(n, 0)^2:
seq(a(n), n=0..16);  # Alois P. Heinz, Aug 12 2025

Mathematica

Table[(PolyLog[ -z, 1/2]/2)^2, {z, 1, 16}] (* Elizabeth A. Blickley (Elizabeth.Blickley(AT)gmail.com), Oct 10 2006 *)

PARI

{a(n)=sum(k=0, n, stirling(n, k, 2)*k!)^2} \\ Paul D. Hanna, Nov 07 2009

Formula

a(n) = Sum_{m>=0} Sum_{k>=0} ((k*m)^n/2^(k+m+2)).

G.f.: Sum_{n>=0} (1/(2-exp(n*x))/2^(n+1)).

Sum_{n>=0} a(n)*log(1+x)^n/n! = o.g.f. of A101370. - Paul D. Hanna, Nov 07 2009

a(n) ~ (n!)^2 / (4 * (log(2))^(2*n+2)). - Vaclav Kotesovec, May 03 2015

Extensions

More terms from Elizabeth A. Blickley (Elizabeth.Blickley(AT)gmail.com), Oct 10 2006