A117487 - OEIS

%I #18 Sep 08 2022 08:45:24

%S 1,2,5,10,20,36,63,104,169,264,405,604,888,1278,1815,2536,3502,4772,

%T 6437,8586,11352,14866,19315,24890,31851,40466,51089,64092,79952,

%U 99172,122386,150264,183639,223394,270605,326422,392225,469490,559970,665542,788412

%N G.f.: 1/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^5))^2.

%C Molien series for S_5 X S_5, cf. A001401.

%C Molien series for S_k X S_k approaches A000712 as k increases.

%C Column 5 of table A115994.

%C Note that a(5) is 20, the scalar product of (1 1 2 3 5) and (5 3 2 1 1 ). a(6) is 36, the scalar product of (1 1 2 3 5 7) and (7 5 3 2 1 1 ).

%H G. C. Greubel, <a href="/A117487/b117487.txt">Table of n, a(n) for n = 1..1000</a>

%p # adapted from A115994 kmax := 120 : qmax := kmax/2 : g:=sum(t^k*q^(k^2)/product((1-q^j)^2, j=1..k), k=1..kmax): gser:=series(g, q=0, qmax): for n from 25 to qmax-1 do P :=coeff(gser, q^n) : printf("%a,",coeff(P, t^5)); od: # _R. J. Mathar_, Apr 07 2006

%t CoefficientList[Series[1/(Product[(1-x^j), {j,5}])^2, {x,0,45}], x] (* _G. C. Greubel_, Jan 01 2020 *)

%o (Magma) n:=5; G:=SymmetricGroup(n); H:=DirectProduct(G,G); MolienSeries(H); // _N. J. A. Sloane_

%o (PARI) my(x='x+O('x^45)); Vec( 1/(prod(j=1,5, 1-x^j))^2 ) \\ _G. C. Greubel_, Jan 01 2020

%o (Sage)

%o def A117487_list(prec):

%o     P.<x> = PowerSeriesRing(ZZ, prec)

%o     return P( 1/(product(1-x^j for j in (1..5)))^2 ).list()

%o A117487_list(45) # _G. C. Greubel_, Jan 01 2020

%Y Cf. A000027, A000712, A001399, A001400, A006918.

%Y Cf. A115994, A117485, A117486, A117488, A117489.

%K nonn

%O 1,2

%A _Alford Arnold_, Mar 22 2006

%E More terms from _R. J. Mathar_, Apr 07 2006

%E Entry revised by _N. J. A. Sloane_, Mar 10 2007