login
This site is supported by donations to The OEIS Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A048163 a(n) = Sum_{k=1..n} ((k-1)!)^2*Stirling2(n,k)^2. 10
1, 2, 14, 230, 6902, 329462, 22934774, 2193664790, 276054834902, 44222780245622, 8787513806478134, 2121181056663291350, 611373265185174628502, 207391326125004608457782, 81791647413265571604175094, 37109390748309009878392597910 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Chad Brewbaker, A combinatorial interpretation of the poly-Bernoulli numbers and two Fermat analogues, INTEGERS Vol. 8 (2008), #A02.

Hyeong-Kwan Ju and Seunghyun Seo, Enumeration of (0,1)-matrices avoiding some 2 X 2 matrices, Discrete Math., 312 (2012), 2473-2481.

Ken Kamano, Lonesum decomposable matrices, arXiv:1701.07157 [math.CO], 2017.

H.-K. Kim et al., Poly-Bernoulli numbers and lonesum matrices, arXiv:1103.4884 [math.CO], 2011.

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

a(n+1) = Sum_{k = 0..n} A163626(n,k)^2. - Philippe Deléham, May 30 2015

a(n) = A306209(2n-2,n-1). - Alois P. Heinz, Feb 01 2019

EXAMPLE

1

1 + 1 = 2

1 + 9 + 4 = 14

1 + 49 + 144 + 36 = 230

1 + 225 + 2500 + 3600 + 576 = 6902

... - Philippe Deléham, May 30 2015

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))

for(n=1, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Jan 06 2013

(PARI) a(n) = sum(k=1, n, (k-1)!^2*stirling(n, k, 2)^2); \\ Michel Marcus, Jun 22 2018

CROSSREFS

Main diagonal of array A099594.

Cf. A220181, A306209.

Sequence in context: A251692 A323693 A118086 * A093548 A052215 A053846

Adjacent sequences:  A048160 A048161 A048162 * A048164 A048165 A048166

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, R. K. Guy

EXTENSIONS

Entry revised by N. J. A. Sloane, Jul 05 2012

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 7 22:29 EST 2019. Contains 329850 sequences. (Running on oeis4.)