|
|
A048163
|
|
a(n) = Sum_{k=1..n} ((k-1)!)^2*Stirling2(n,k)^2.
|
|
12
|
|
|
1, 2, 14, 230, 6902, 329462, 22934774, 2193664790, 276054834902, 44222780245622, 8787513806478134, 2121181056663291350, 611373265185174628502, 207391326125004608457782, 81791647413265571604175094, 37109390748309009878392597910, 19192672725746588045912535407702
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
a(n) is also the number of max-closed relations on an ordered n-element domain (see the paper by Jeavons and Cooper, 1995). - Don Knuth, Feb 12 2024
|
|
REFERENCES
|
Lovasz, L. and Vesztergombi, K.; Restricted permutations and Stirling numbers. Combinatorics (Proc. Fifth Hungarian Colloq., Keszthely, 1976), Vol. II, pp. 731-738, Colloq. Math. Soc. Janos Bolyai, 18, North-Holland, Amsterdam-New York, 1978.
K. Vesztergombi, Permutations with restriction of middle strength, Stud. Sci. Math. Hungar., 9 (1974), 181-185.
|
|
LINKS
|
|
|
FORMULA
|
E.g.f. (with offset 0): Sum((1-exp(-(m+1)*z))^m, m=0..oo)
O.g.f.: Sum_{n>=1} n^(n-1) * (n-1)! * x^n / Product_{k=1..n-1} (1 - n*k*x). - Paul D. Hanna, Jan 05 2013
Limit n->infinity (a(n)/n!)^(1/n)/n = 1/(exp(1)*(log(2))^2) = 0.7656928576... . - Vaclav Kotesovec, Jun 21 2013
a(n) ~ 2*sqrt(Pi) * n^(2*n-3/2) / (sqrt(1-log(2)) * exp(2*n) * (log(2))^(2*n-1)). - Vaclav Kotesovec, May 13 2014
|
|
EXAMPLE
|
1
1 + 1 = 2
1 + 9 + 4 = 14
1 + 49 + 144 + 36 = 230
1 + 225 + 2500 + 3600 + 576 = 6902
|
|
MATHEMATICA
|
Table[Sum[((k-1)!)^2*StirlingS2[n, k]^2, {k, 1, n}], {n, 1, 20}] (* Vaclav Kotesovec, Jun 21 2013 *)
|
|
PROG
|
(PARI) a(n)=if(n<1, 0, polcoeff(sum(m=1, n, m^(m-1)*(m-1)!*x^m/prod(k=1, m-1, 1+m*k*x+x*O(x^n))), n)) \\ Paul D. Hanna, Jan 05 2013
for(n=1, 20, print1(a(n), ", "))
(PARI) Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)
a(n)=sum(k=1, n, (-1)^(n-k)*k^(n-1)*(k-1)!*Stirling2(n-1, k-1))
(PARI) a(n) = sum(k=1, n, (k-1)!^2*stirling(n, k, 2)^2); \\ Michel Marcus, Jun 22 2018
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,changed
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|