A136246 a(n) = (1/n!)*Sum_{k=0..n} (-1)^(n-k)*Stirling1(n,k)*A062208(k).
1, 1, 32, 2712, 449102, 122886128, 50225389432, 28670796914144, 21789885975738524, 21271115441652577064, 25938193213744579451420, 38638907727108476424404864, 69044758685363149615280762608, 145768622491129079115419544343808, 358961215083489204505055286181798208
Offset: 0
Links
- Andrew Howroyd, Table of n, a(n) for n = 0..100
Programs
-
Maple
A000629 := proc(n) local k ; sum( k^n/2^k,k=0..infinity) ; end: A062208 := proc(n) option remember ; local a,stir,ni,n1,n2,n3,stir2,i,j,tmp ; a := 0 ; if n = 0 then RETURN(1) ; fi ; stir := combinat[partition](n) ; stir2 := {} ; for i in stir do if nops(i) <= 3 then tmp := i ; while nops(tmp) < 3 do tmp := [op(tmp),0] ; od: tmp := combinat[permute](tmp) ; for j in tmp do stir2 := stir2 union { j } ; od: fi ; od: for ni in stir2 do n1 := op(1,ni) ; n2 := op(2,ni) ; n3 := op(3,ni) ; a := a+combinat[multinomial](n,n1,n2,n3)*(A000629(3*n1+2*n2+n3)-1/2-2^(3*n1+2*n2+n3)/4)*(-3)^n2*2^n3 ; od: a/(2*6^n) ; end: A136246 := proc(n) local k ; add((-1)^(n-k)*combinat[stirling1](n,k)*A062208(k),k=0..n)/n! ; end: seq(A136246(n),n=0..14) ; # R. J. Mathar, Apr 01 2008
-
Mathematica
a[n_] := Sum[Binomial[Binomial[j, 3] + n - 1, n] * Sum[(-1)^(i - j)* Binomial[i, j], {i, j, 3n}], {j, 0, 3n}]; a /@ Range[0, 14] (* Jean-François Alcover, Feb 13 2020, after Andrew Howroyd *) -
PARI
a(n) = {sum(j=0, 3*n, binomial(binomial(j,3)+n-1, n) * sum(i=j, 3*n, (-1)^(i-j)*binomial(i,j)))} \\ Andrew Howroyd, Feb 09 2020
Formula
a(n) = Sum_{m>=0} binomial(binomial(m,3)+n-1,n)/2^(m+1).
a(n) = Sum_{j=0..3*n} binomial(binomial(j,3)+n-1, n) * (Sum_{i=j..3*n} (-1)^(i-j)*binomial(i,j)). - Andrew Howroyd, Feb 09 2020
Extensions
More terms from R. J. Mathar, Apr 01 2008
Terms a(13) and beyond from Andrew Howroyd, Feb 09 2020