A163845 Row sums of triangle A163842.

1, 13, 109, 765, 4881, 29369, 169919, 956237, 5272945, 28632525, 153638211, 816715073, 4309138419, 22598433555, 117926579385, 612863125965, 3174156512865, 16392351740045, 84448387609475, 434142126555125, 2227861180841895, 11414655603043335, 58403793025471605

Offset: 0

Author

Peter Luschny, Aug 06 2009

Keywords

• nonn
• changed

Links

• Peter Luschny, Swinging Factorial.

Crossrefs

Cf. A056040, A163842.

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(2*i+1),i=k..n),k=0..n) end:

Mathematica

swing[n_] := n! / Floor[n/2]!^2; a[n_] := Sum[Binomial[n-k, n-i] * swing[2*i+1], {k, 0, n}, {i, k, n}]; Array[a, 30, 0] (* Amiram Eldar, Aug 22 2025 *)

Formula

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