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

1,4

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

Links

Table of n, a(n) for n=1..23.

Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets

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

Example

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.

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 *)

Prog

(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 */

Crossrefs

Cf. A001563, A057438, A122974.

Sequence in context: A379281 A152392 A001563 * A094258 A234854 A334735

Adjacent sequences: A094301 A094302 A094303 * A094305 A094306 A094307

Keyword

nonn

Author

Amarnath Murthy, Apr 29 2004

Extensions

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

Status

approved