A002775 a(n) = n^2 * n!.

0, 1, 8, 54, 384, 3000, 25920, 246960, 2580480, 29393280, 362880000, 4829932800, 68976230400, 1052366515200, 17086945075200, 294226732800000, 5356234211328000, 102793666719744000, 2074369080655872000, 43913881247588352000, 973160803270656000000, 22531105497723863040000

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Denominators in power series expansion of the higher order exponential integral E(x,m=2,n=1) - (gamma^2/2 + Pi^2/12 + gamma*log(x) + log(x)^2/2), n>0, see A163931. - Johannes W. Meijer, Oct 16 2009

References

• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Delbert L. Johnson, Table of n, a(n) for n = 0..447
• J. D. H. Dickson, Discussion of two double series arising from the number of terms in determinants of certain forms, Proc. London Math. Soc., 10 (1879), 120-122.
• J. D. H. Dickson, Discussion of two double series arising from the number of terms in determinants of certain forms, Proc. London Math. Soc., 10 (1879), 120-122. [Annotated scanned copy]

Crossrefs

Cf. A000290, A000142, A047922.

Cf. A163931 (E(x,m,n)), A001563 (n*n!), A091363 (n^3*n!), A091364 (n^4*n!). - Johannes W. Meijer, Oct 16 2009

Cf. A367731, A367732.

Programs

Maple

with(combinat):for n from 0 to 15 do printf(`%d, `,n!/2*sum(2*n, k=1..n)) od: # Zerinvary Lajos, Mar 13 2007
seq(sum(sum(mul(k, k=1..n),l=1..n),m=1..n), n=0..21); # Zerinvary Lajos, Jan 26 2008
with (combstruct):a:=proc(m) [ZL, {ZL=Set(Cycle(Z, card>=m))}, labeled]; end: ZLL:=a(1):seq(count(ZLL, size=n)*n^2, n=0..21); # Zerinvary Lajos, Jun 11 2008
a:=n->add(0+add(n!, j=1..n),j=1..n):seq(a(n), n=0..21); # Zerinvary Lajos, Aug 27 2008

Mathematica

nn=20;a=1/(1-x); Range[0,nn]! CoefficientList[Series[x D[x D[a,x], x], {x,0,nn}], x] (* Geoffrey Critzer, Jan 17 2012 *)
Table[n^2 n!,{n,0,40}] (* Harvey P. Dale, Aug 01 2021 *)

Formula

E.g.f.: x*(1+x)/(1-x)^3. - Vladeta Jovovic, Dec 01 2002

E.g.f.: x*A'(x) where A(x) is the e.g.f. for A001563. - Geoffrey Critzer, Jan 17 2012

From Alexander Adamchuk, Oct 24 2004: (Start)

Sum of all matrix elements M(i, j) = i/(i+j) multiplied by 2*n!.

a(n) = 2 * n! * Sum_{j=1..n} Sum_{i=1..n} i/(i+j).

Example: a(2) = 2*2! * (1/(1+1) + 1/(1+2) + 2/(2+1) + 2/(2+2)) = 8. (End)

From Amiram Eldar, Dec 24 2023: (Start)

Sum_{n>=1} 1/a(n) = A367731.

Sum_{n>=1} (-1)^(n+1)/a(n) = A367732. (End)