A094304 Sum of all possible sums formed from all but one of the previous terms, starting 1.

1, 0, 1, 4, 18, 96, 600, 4320, 35280, 322560, 3265920, 36288000, 439084800, 5748019200, 80951270400, 1220496076800, 19615115520000, 334764638208000, 6046686277632000, 115242726703104000, 2311256907767808000, 48658040163532800000, 1072909785605898240000

Offset: 1

Author

Amarnath Murthy, Apr 29 2004

Keywords

• nonn

Comments

Apart from initial 1, same sequence as A001563. Additive analog of A057438.

a(1) = 1, for n >= 2: a(n) = sum of previous terms * (n-2) = (Sum_(i=1...n-2) a(i)) * (n-2). a(n) = A001563(n-2) = A094258(n-1) for n >= 3. - Jaroslav Krizek, Oct 16 2009

Examples

			a(2) = 0 as there is only one previous term and empty sum is taken to be 0.
a(4) = (a(1) +a(2))+ (a(1) +a(3)) + (a(2) +a(3)) = (1+0) +(1+1) +(0+1) = 4.
a(5) = (a(1)+a(2)+a(3)) +(a(1)+a(2)+ a(4)) +(a(1)+a(3)+a(4)) +(a(2)+a(3)+a(4)) = (1+0+1) +(1+0+4) +(1+1+4) +(0+1+4) = 2 + 5 + 6 + 5 = 18.
		

Links

• Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets

Crossrefs

Cf. A001563, A057438, A122974.

Programs

Maple

a := n -> (n-2)*(n-2)!: 1,seq(a(n), n=2..23); # Emeric Deutsch, May 01 2008

Mathematica

In[2]:= l = {1}; Do[k = Length[l] - 1; p = Plus @@ Flatten[Select[Subsets[l], Length[ # ]==k& ]]; AppendTo[l, p], {n, 20}]; l (* Ryan Propper, May 28 2006 *)

PARI

v=vector(30);v[1]=1;v[2]=0;for(n=3,#v,s=0;for(i=1,2^(n-1)-1, vb=binary(i); if(hammingweight(vb)==n-2,s=s+sum(j=1,#vb, if(vb[j], v[n-#vb+j-1]))));v[n]=s;print1(s,",")) /* Ralf Stephan, Sep 22 2013 */

Formula

a(n) = (n-2)!(n-2) for n>=2. - Emeric Deutsch, May 01 2008

G.f.: x*T(0), where T(k) = 1 - x^2*(k+1)^2/(x^2*(k+1)^2 - (1 -x -2*x*k)*(1 -3*x -2*x*k)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 10 2013

a(n) = S1(n,1) - S1(n-1,1), where S1 are the unsigned Stirling cycle numbers. - Peter Luschny, Apr 10 2016

a(n) = A122974(n-1,n-1). - Alois P. Heinz, Nov 24 2019

Extensions

Edited by N. J. A. Sloane, May 29 2006