OFFSET
0,3
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..100
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
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 *)
PROG
(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
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Vladeta Jovovic, Mar 16 2008
EXTENSIONS
More terms from R. J. Mathar, Apr 01 2008
Terms a(13) and beyond from Andrew Howroyd, Feb 09 2020
STATUS
approved