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URL: https://oeis.org/A163845

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A163845
Row sums of triangle A163842.
1
1, 13, 109, 765, 4881, 29369, 169919, 956237, 5272945, 28632525, 153638211, 816715073, 4309138419, 22598433555, 117926579385, 612863125965, 3174156512865, 16392351740045, 84448387609475, 434142126555125, 2227861180841895, 11414655603043335, 58403793025471605
OFFSET
0,2
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).
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 *)
CROSSREFS
Sequence in context: A055840 A243417 A367592 * A380001 A075143 A005769
KEYWORD
nonn
AUTHOR
Peter Luschny, Aug 06 2009
STATUS
approved