A163774 Row sums of the central coefficients triangle (A163771).

1, 3, 13, 51, 201, 783, 3039, 11763, 45481, 175803, 679779, 2630367, 10187659, 39500373, 153329913, 595883763, 2318471289, 9030982491, 35216266947, 137469149451, 537152523711, 2100857828193, 8223917499477, 32219655346719, 126328429601451, 495676719721953, 1946227355491909

Offset: 0

Author

Peter Luschny, Aug 05 2009

Keywords

• nonn
• changed

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.

Crossrefs

Cf. A056040, A163771.

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((-1)^(n-i)*binomial(n-k,n-i)*swing(2*i),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], {i, k, n}]; Table[Sum[t[n, k], {k, 0, n}], {n, 0, 50}] (* G. C. Greubel, Aug 04 2017 *)

Formula

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

Conjecture: a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n+1,k)*binomial(2*k,k). - Werner Schulte, Nov 17 2015

Extensions

More terms from Michel Marcus, Nov 24 2015