A163843 Row sums of triangle A163840.

1, 3, 10, 41, 116, 427, 1240, 4181, 12472, 40091, 121364, 380701, 1160186, 3593969, 10979532, 33785469, 103258800, 316532947, 966976444, 2957131673, 9026437602, 27558146133, 84043120308, 256263107177, 780817641926

Offset: 0

Author

Peter Luschny, Aug 06 2009

Keywords

• nonn
• changed

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Peter Luschny, Swinging Factorial.

Crossrefs

Cf. A056040, A163840.

Programs

Maple

swing := proc(n) option remember; if n = 0 then 1 elif
irem(n, 2) = 1 then swing(n-1)*n else 4*swing(n-1)/n fi end:
a := proc(n) local i,k; add(add(binomial(n-k,n-i)*swing(i),i=k..n),k=0..n) end:

Mathematica

sf[n_] := n!/Quotient[n, 2]!^2; t[n_, k_] := Sum[Binomial[n - k, n - i]*sf[i], {i, k, n}]; Table[Sum[t[n, k], {k, 0, n}], {n, 0, 50}] (* G. C. Greubel, Aug 06 2017 *)

Formula

a(n) = Sum_{k=0..n} Sum_{i=k..n} binomial(n-k,n-i)*i$ where i$ denotes the swinging factorial of i (A056040).