A283322 - OEIS

%I #13 Mar 16 2017 15:51:23

%S 1,2,4,22,149,1186,10807,110762,1260289,15757714,214703831,3165856882,

%T 50220944017,852735163034,15429720661279,296365775922106,

%U 6021825238479617,129039801791351842,2908148713706872999,68758376703814729154,1701649010958291917521,43990236798804135274282

%N Row sums of triangle in A283321.

%H Indranil Ghosh, <a href="/A283322/b283322.txt">Table of n, a(n) for n = 0..400</a>

%H G. N. Bakare, S. O. Makanjuola, <a href="http://kwsman.com/articles/Revised%20Bakare.pdf">Some Results on Properties of Alternating Semigroups</a>, Nigerian Journal of Mathematics and Applications Volume 24,(2015), 184-192.

%F Bakare et al. give a formula, see Theorem 3.2.

%e Row 3 of triangle A283321: 1, 3, 3, 9. So a(3) = 1 + 3 + 3 + 9 = 22. - _Indranil Ghosh_, Mar 16 2017

%t T[n_, k_]:=If[k==n, (n !/2), If[k==n - 1, n^2*(n - 1)!/2, Binomial[n,k]^2 * k !]]; t[n_,k_]:=If[n<2, 1, T[n, k]]; For[n=0, n<=20, Print[Sum[t[n, k], {k, 0, n}]," "]; n++] (* _Indranil Ghosh_, Mar 16 2017 *)

%o (PARI) T(n,k) = if(k==n, (n!/2), if(k==n - 1, n^2*(n - 1)!/2,  binomial(n, k)^2 * k!));

%o t(n,k) = if(n<2, 1, T(n, k));

%o {for(n=0, 21, print1(sum(k=0, n, t(n,k)),", "))} \\ _Indranil Ghosh_, Mar 16 2017

%o (Python)

%o import math

%o f=math.factorial

%o def C(n,r): return f(n)/f(r)/f(n - r)

%o def T(n,k):

%o ....if k==n: return f(n)/2

%o ....elif k==n-1: return n**2 * f(n - 1) / 2

%o ....else: return C(n, k)**2 * f(k)

%o i=0

%o l=[]

%o for n in range(0,401):

%o ....for k in range(0, n+1):

%o ........if n<2: l+=[1,]

%o ........else: l+=[T(n,k),]

%o ....print str(i)+" "+str(sum(l))

%o ....l=[]

%o ....i+=1 # _Indranil Ghosh_, Mar 16 2017

%Y Cf. A283321.

%K nonn

%O 0,2

%A _N. J. A. Sloane_, Mar 15 2017