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proposed

a(n) = Sum_{k=0..n} Sum_{i=k..n} (-1)^(n-i)*binomial(n-k,n-i)*i$

where i$ denotes the swinging factorial of i (A056040).

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

proposed

editing

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proposed

1, 1, 4, 15, -14, 185, -454, 2107, -6194, 22689, -70058, 234971, -734304, 2368379, -7404318, 23417955, -72988938, 228324569, -708982738, 2202742447, -6815736144, 21077285943, -65016664062, 200371842727, -616463969324, 1894794918275, -5816606133674, 17839764136377

a(n) = sumSum_{k=0..n} sumSum_{i=k..n} (-1)^(n-i)*binomial(n-k,n-i)*i$

G. C. Greubel, <a href="/A163773/b163773.txt">Table of n, a(n) for n = 0..1000</a>

sf[n_] := n!/Quotient[n, 2]!^2; t[n_, k_] := Sum[(-1)^(n - i)*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 03 2017 *)

Cf. A163770.

Terms a(18) onward added by G. C. Greubel, Aug 03 2017

approved

editing

_Peter Luschny (peter(AT)luschny.de), _, Aug 05 2009

Row sums of the swinging derangement triangle (A163770).

1, 1, 4, 15, -14, 185, -454, 2107, -6194, 22689, -70058, 234971, -734304, 2368379, -7404318, 23417955, -72988938, 228324569

0,3

a(n) = sum{k=0..n} sum{i=k..n} (-1)^(n-i)*binomial(n-k,n-i)*i$

where i$ denotes the swinging factorial of i (A056040).

Peter Luschny, <a href="http://www.luschny.de/math/swing/SwingingFactorial.html"> Swinging Factorial</a>.

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(i), i=k..n), k=0..n) end:

Cf. A163770

sign

Peter Luschny (peter(AT)luschny.de), Aug 05 2009

approved