A105219 a(n) = Sum_{k=0..n} C(n,k)^2*(n-k)!*k^2.

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

Vincenzo Librandi, Table of n, a(n) for n = 0..200

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

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

Sequence in context: A001090 A243782 A369810 * A060071 A037205 A302399

Adjacent sequences: A105216 A105217 A105218 * A105220 A105221 A105222

Keyword

easy,nonn

Author

Miklos Kristof, Apr 13 2005

Status

approved