A105219 - OEIS

%I #48 Feb 23 2022 17:29:50

%S 0,1,8,63,544,5225,55656,653023,8379008,116780049,1757211400,

%T 28394129951,490371506208,9013522796473,175679564492264,

%U 3618800515187775,78547755741723136,1791704327280481313,42846080320725932808,1071798626271975328639,27989931083161219661600

%N a(n) = Sum_{k=0..n} C(n,k)^2*(n-k)!*k^2.

%C 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.)

%C 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

%H Vincenzo Librandi, <a href="/A105219/b105219.txt">Table of n, a(n) for n = 0..200</a>

%H John Riordan, <a href="/A002720/a002720_3.pdf">Letter to N. J. A. Sloane, Sep 26 1980 with notes on the 1973 Handbook of Integer Sequences</a>. Note that the sequences are identified by their N-numbers, not their A-numbers.

%F E.g.f.: (x/(1-x)^2+x^2/(1-x)^3)*exp(x/(1-x)).

%F a(n) = n^2*A002720(n-1) for n>=1 [Riordan]. - _N. J. A. Sloane_, Jan 10 2018

%F 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

%F 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

%F 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

%F 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

%F a(n) = Sum_{k=0...n} A144084(n,k)*k. - _Geoffrey Critzer_, Nov 17 2021

%F a(n) = Sum_{k=0..n} (n-k) * A206703(n,k). - _Alois P. Heinz_, Feb 19 2022

%e b(n) = 0,1,4,9,16,25,36,49,64,...

%e 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.

%p 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);

%p seq(`if`(n=0,0,simplify(n!*LaguerreL(n-1,2,-1))),n=0..17); # _Peter Luschny_, Apr 11 2015

%t 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 *)

%Y Cf. A000290, A002720, A144084, A202410, A206703.

%K easy,nonn

%O 0,3

%A _Miklos Kristof_, Apr 13 2005