A163775 Row sums of triangle A163772.

1, 11, 73, 403, 2021, 9567, 43611, 193683, 844213, 3629083, 15437951, 65143503, 273148279, 1139548469, 4734740493, 19606960755, 80969809797, 333601494651

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.

Peter Luschny, Swinging Factorial.

Formula

a(n) = Sum_{k=0..n} Sum_{i=k..n} (-1)^(n-i)*binomial(n-k,n-i)*(2i+1)$ where i$ denotes the swinging factorial of i (A056040).

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((-1)^(n-i)*binomial(n-k, n-i)*swing(2*i+1), i=k..n), k=0..n) end:

Mathematica

sf[n_] := n!/Quotient[n, 2]!^2; t[n_, k_] := Sum[(-1)^(n - i)* Binomial[n - k, n - i]*sf[2*i + 1], {i, k, n}]; Table[Sum[t[n, k], {k, 0, n}], {n, 0, 50}] (* G. C. Greubel, Aug 04 2017 *)

Crossrefs

Cf. A163772.

Sequence in context: A123039 A226034 A217946 * A092244 A342830 A155634

Adjacent sequences: A163772 A163773 A163774 * A163776 A163777 A163778

Keyword

nonn

Author

Peter Luschny, Aug 05 2009

Status

approved