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
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
Programs
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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
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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)
Comments