OFFSET
0,3
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]
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)
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 *)
CROSSREFS
Cf. A163931 (E(x,m,n)), A001563 (n*n!), A091363 (n^3*n!), A091364 (n^4*n!). - Johannes W. Meijer, Oct 16 2009
KEYWORD
nonn,easy
AUTHOR
STATUS
approved