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A105219
a(n) = Sum_{k=0..n} C(n,k)^2*(n-k)!*k^2.
6
0, 1, 8, 63, 544, 5225, 55656, 653023, 8379008, 116780049, 1757211400, 28394129951, 490371506208, 9013522796473, 175679564492264, 3618800515187775, 78547755741723136, 1791704327280481313, 42846080320725932808, 1071798626271975328639, 27989931083161219661600
OFFSET
0,3
COMMENTS
If the e.g.f. of n^2 is E(x) and a(n) = Sum_{k=0..n} C(n,k)^2*(n-k)!*k^2, then the e.g.f. of a(n) is E(x/(1-x))/(1-x). (Thanks to Vladeta Jovovic for help.)
a(n) is the total number of edges in all matchings of the labeled complete bipartite graph K_n,n. Cf. A144084 for other interpretations. - Geoffrey Critzer, Nov 17 2021
LINKS
John Riordan, Letter to N. J. A. Sloane, Sep 26 1980 with notes on the 1973 Handbook of Integer Sequences. Note that the sequences are identified by their N-numbers, not their A-numbers.
FORMULA
E.g.f.: (x/(1-x)^2+x^2/(1-x)^3)*exp(x/(1-x)).
a(n) = n^2*A002720(n-1) for n>=1 [Riordan]. - N. J. A. Sloane, Jan 10 2018
a(n) = (n+1)!*(2*L(n,-1)-L(n+1,-1)) where L(n,x) is the n-th Laguerre polynomial. - Peter Luschny, Jan 19 2012
Recurrence: a(n) = 2*(n+2)*a(n-1) - (n^2+4*n-4)*a(n-2) + 2*(n-2)*(n-1)*a(n-3). - Vaclav Kotesovec, Oct 17 2012
a(n) ~ exp(2*sqrt(n)-n-1/2)*n^(n+5/4)/sqrt(2)*(1-17/(48*sqrt(n))). - Vaclav Kotesovec, Oct 17 2012
a(n) = n!*L(n-1,2,-1) for n>=1 where L(n,b,x) is the n-th generalized Laguerre polynomial. - Peter Luschny, Apr 11 2015
a(n) = Sum_{k=0...n} A144084(n,k)*k. - Geoffrey Critzer, Nov 17 2021
a(n) = Sum_{k=0..n} (n-k) * A206703(n,k). - Alois P. Heinz, Feb 19 2022
EXAMPLE
b(n) = 0,1,4,9,16,25,36,49,64,...
a(3) = C(3,0)^2*3!*b(0) + C(3,1)^2*2!*b(1) + C(3,2)^2*1!*b(2) + C(3,3)^2*0!*b(3) = 1*6*0 + 9*2*1 + 9*1*4 + 1*1*9 = 0 + 18 + 36 + 9 = 63.
MAPLE
for n from 0 to 30 do b[n]:=n^2 od: seq(add(binomial(n, k)^2*(n-k)!*b[k], k=0..n), n=0..30);
seq(`if`(n=0, 0, simplify(n!*LaguerreL(n-1, 2, -1))), n=0..17); # Peter Luschny, Apr 11 2015
MATHEMATICA
CoefficientList[Series[(x/(1-x)^2+x^2/(1-x)^3)*E^(x/(1-x)), {x, 0, 20}], x]* Table[n!, {n, 0, 20}] (* Vaclav Kotesovec, Oct 17 2012 *)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Miklos Kristof, Apr 13 2005
STATUS
approved